Perturbation Of The Form H ′ = I Λ [ A , H 0 ] H'=i\lambda[A,H_0] H ′ = Iλ [ A , H 0 ​ ]

$$

Introduction to Perturbation Theory

Perturbation theory is a fundamental concept in quantum mechanics that allows us to study the behavior of a quantum system that is subject to a small external perturbation. The perturbation can be thought of as a small disturbance that affects the energy levels and wavefunctions of the system. In this article, we will discuss a specific type of perturbation of the form $H'=i\lambda[A,H_0]$, where $H_0$ is the unperturbed Hamiltonian, $A$ is an operator, and $\lambda$ is a small parameter.

Recall of Perturbation Theory

Before we dive into the specifics of the perturbation $H'=i\lambda[A,H_0]$, let's recall the correction to the wavefunction up to first order. The correction to the wavefunction is given by:

$$\psi_n=\psi_n^{(0)}+\sum_{m\neq n}\frac{\langle\psi_m^0|H_1|\psi_n^0\rangle}{E_n^0-E_m^0}\psi_m^{(0)}$$

where $\psi_n^{(0)}$ is the unperturbed wavefunction, $H_1$ is the perturbation Hamiltonian, and $E_n^0$ is the unperturbed energy level.

Perturbation of the Form $H'=i\lambda[A,H_0]$

Now, let's consider the perturbation of the form $H'=i\lambda[A,H_0]$. This perturbation can be thought of as a small rotation of the system, where the operator $A$ represents the rotation axis and the parameter $\lambda$ represents the magnitude of the rotation.

To study the effect of this perturbation, we need to calculate the correction to the wavefunction up to first order. Using the formula above, we get:

$$\psi_n=\psi_n^{(0)}+\sum_{m\neq n}\frac{\langle\psi_m^0|i\lambda[A,H_0]|\psi_n^0\rangle}{E_n^0-E_m^0}\psi_m^{(0)}$$

Evaluation of the Matrix Element

To evaluate the matrix element $\langle\psi_m^0|i\lambda[A,H_0]|\psi_n^0\rangle$, we need to use the properties of the operators $A$ and $H_0$. Since $A$ is an operator that represents a rotation, we can assume that it commutes with the unperturbed Hamiltonian $H_0$. Therefore, we can write:

$$[A,H_0]=0$$

Using this result, we can simplify the matrix element as follows:

$$\langle\psi_m^0|i\lambda[A,H_0]|\psi_n^0\rangle=i\lambda\langle\psi_m^0|A[H_0,\psi_n^0]\rangle$$

Evaluation of the Commutator

To evaluate the commutator $[H_0,\psi_n^0]$, we need to use the properties of the unperturbed Hamiltonian $H_0$. Since $H_0$ is a Hermitian operator, we can write:

$$[H_0,\psi_n^0]=i\hbar\frac{\partial\psi_n^0}{\partial t}$$

Using this result, we can simplify the matrix element as follows:

$$\langle\psi_m^0|i\lambda[A,H_0]|\psi_n^0\rangle=-\lambda\hbar\langle\psi_m^0|A\frac{\partial\psi_n^0}{\partial t}\rangle$$

Evaluation of the Time Derivative

To evaluate the time derivative $\frac{\partial\psi_n^0}{\partial t}$, we need to use the time-dependent Schrödinger equation:

$$i\hbar\frac{\partial\psi_n^0}{\partial t}=H_0\psi_n^0$$

Using this result, we can simplify the matrix element as follows:

$$\langle\psi_m^0|i\lambda[A,H_0]|\psi_n^0\rangle=\lambda\hbar^2\langle\psi_m^0|A\frac{H_0\psi_n^0}{E_n^0}\rangle$$

Simplification of the Matrix Element

Using the properties of the operators $A$ and $H_0$, we can simplify the matrix element as follows:

$$\langle\psi_m^0|i\lambda[A,H_0]|\psi_n^0\rangle=\lambda\hbar^2\langle\psi_m^0|AH_0\psi_n^0\rangle$$

Evaluation of the Matrix Element

To evaluate the matrix element $\langle\psi_m^0|AH_0\psi_n^0\rangle$, we need to use the properties of the operators $A$ and $H_0$. Since $A$ is an operator that represents a rotation, we can assume that it commutes with the unperturbed Hamiltonian $H_0$. Therefore, we can write:

$$\langle\psi_m^0|AH_0\psi_n^0\rangle=\langle\psi_m^0|H_0A\psi_n^0\rangle$$

Using this result, we can simplify the matrix element as follows:

$$\langle\psi_m^0|i\lambda[A,H_0]|\psi_n^0\rangle=\lambda\hbar^2\langle\psi_m^0|H_0A\psi_n^0\rangle$$

Conclusion

In this article, we have discussed a specific type of perturbation of the form $H'=i\lambda[A,H_0]$. We have evaluated the correction to the wavefunction up to first order and simplified the matrix element using the properties of the operators $A$ and $H_0$. The result shows that the perturbation causes a small rotation of the system, where the operator $A$ represents the rotation axis and the parameter $\lambda$ represents the magnitude of the rotation.

Future Work

In the future, we plan to study the effect of this perturbation on the energy levels and wavefunctions of the system. We will also investigate the possibility of using this perturbation to control the behavior of the system.

References

• [1] Dirac, P. A. M. (1958). The Principles of Quantum Mechanics. Oxford University Press.
• [2] Landau, L. D., & Lifshitz, E. M. (1977). Quantum Mechanics: Non-Relativistic Theory. Pergamon Press.
• [3] Messiah, A. (1961). Quantum Mechanics. John Wiley & Sons.

Appendix

Evaluation of the Commutator

To evaluate the commutator $[H_0,\psi_n^0]$, we need to use the properties of the unperturbed Hamiltonian $H_0$. Since $H_0$ is a Hermitian operator, we can write:

$$[H_0,\psi_n^0]=i\hbar\frac{\partial\psi_n^0}{\partial t}$$

Using this result, we can simplify the matrix element as follows:

$$\langle\psi_m^0|i\lambda[A,H_0]|\psi_n^0\rangle=-\lambda\hbar\langle\psi_m^0|A\frac{\partial\psi_n^0}{\partial t}\rangle$$

Evaluation of the Time Derivative

To evaluate the time derivative $\frac{\partial\psi_n^0}{\partial t}$, we need to use the time-dependent Schrödinger equation:

$$i\hbar\frac{\partial\psi_n^0}{\partial t}=H_0\psi_n^0$$

Using this result, we can simplify the matrix element as follows:

$$\langle\psi_m^0|i\lambda[A,H_0]|\psi_n^0\rangle=\lambda\hbar^2\langle\psi_m^0|A\frac{H_0\psi_n^0}{E_n^0}\rangle$$

Simplification of the Matrix Element

Using the properties of the operators $A$ and $H_0$, we can simplify the matrix element as follows:

$$\langle\psi_m^0|i\lambda[A,H_0]|\psi_n^0\rangle=\lambda\hbar^2\langle\psi_m^0|AH_0\psi_n^0\rangle$$

Evaluation of the Matrix Element

To evaluate the matrix element $\langle\psi_m^0|AH_0\psi_n^0\rangle$, we need to use the properties of the operators $A$ and $H_0$. Since $A$ is an operator that represents a rotation, we can assume that it commutes with the unperturbed Hamiltonian $H_0$. Therefore, we can write:

$$\langle\psi_m^0|AH_0\psi_n^0\rangle=\langle\psi_m^0|H_0A\psi_n^0\rangle$$

Using this result, we can simplify the matrix element as follows:

\langle\psi_m^0|i\lambda[A,H_0]|\psi_n^0\rangle=\lambda\hbar^2\langle\psi_m^0|H_0A\psi_n^0\rangle$<br/> # **Q&A: Perturbation of the Form $H'=i\lambda[A,H_0]$**

Introduction

In our previous article, we discussed a specific type of perturbation of the form H&#x27;=i\lambda[A,H_0]. We evaluated the correction to the wavefunction up to first order and simplified the matrix element using the properties of the operators $A$ and $H_0$. In this article, we will answer some frequently asked questions about this perturbation.

Q: What is the physical meaning of the perturbation H&#x27;=i\lambda[A,H_0]?

A: The perturbation H&#x27;=i\lambda[A,H_0] represents a small rotation of the system, where the operator $A$ represents the rotation axis and the parameter $\lambda$ represents the magnitude of the rotation.

Q: How does the perturbation affect the energy levels of the system?

A: The perturbation causes a small shift in the energy levels of the system. The magnitude of the shift depends on the magnitude of the perturbation parameter $\lambda$ and the energy difference between the unperturbed energy levels.

Q: How does the perturbation affect the wavefunctions of the system?

A: The perturbation causes a small change in the wavefunctions of the system. The magnitude of the change depends on the magnitude of the perturbation parameter $\lambda$ and the overlap between the unperturbed wavefunctions.

Q: Can the perturbation be used to control the behavior of the system?

A: Yes, the perturbation can be used to control the behavior of the system. By adjusting the magnitude of the perturbation parameter $\lambda$, we can control the magnitude of the rotation and the resulting changes in the energy levels and wavefunctions.

Q: What are the limitations of the perturbation H&#x27;=i\lambda[A,H_0]?

A: The perturbation H&#x27;=i\lambda[A,H_0] is only valid for small values of the perturbation parameter $\lambda$. For large values of $\lambda$, the perturbation becomes significant and the system may exhibit non-linear behavior.

Q: Can the perturbation be used in conjunction with other perturbations?

A: Yes, the perturbation H&#x27;=i\lambda[A,H_0] can be used in conjunction with other perturbations. By combining multiple perturbations, we can create more complex and interesting behavior in the system.

Q: How does the perturbation affect the system's dynamics?

A: The perturbation causes a small change in the system's dynamics. The magnitude of the change depends on the magnitude of the perturbation parameter $\lambda$ and the energy difference between the unperturbed energy levels.

Q: Can the perturbation be used to study the system's behavior in different regimes?

A: Yes, the perturbation can be used to study the system's behavior in different regimes. By adjusting the magnitude of the perturbation parameter $\lambda$, we can study the system's behavior in different regimes, such as the classical or quantum regime.

Q: What are the potential applications of the perturbation H&#x27;=i\lambda[A,H_0]?

A: The perturbation H&#x27;=i\lambda[A,H_0] has potential applications in various fields, such as quantum computing, quantum simulation, and quantum control. By using this perturbation, we can create more complex and interesting behavior in the system, which can be used to study various phenomena and develop new technologies.

Conclusion

In this article, we have answered some frequently asked questions about the perturbation H&#x27;=i\lambda[A,H_0]. We have discussed the physical meaning of the perturbation, its effect on the energy levels and wavefunctions of the system, and its potential applications. We hope that this article has provided a useful overview of this perturbation and its potential uses.

References

• [1] Dirac, P. A. M. (1958). The Principles of Quantum Mechanics. Oxford University Press.
• [2] Landau, L. D., & Lifshitz, E. M. (1977). Quantum Mechanics: Non-Relativistic Theory. Pergamon Press.
• [3] Messiah, A. (1961). Quantum Mechanics. John Wiley & Sons.

Appendix

Evaluation of the Commutator

To evaluate the commutator $[H_0,\psi_n^0]$, we need to use the properties of the unperturbed Hamiltonian $H_0$. Since $H_0$ is a Hermitian operator, we can write:

$$[H_0,\psi_n^0]=i\hbar\frac{\partial\psi_n^0}{\partial t} </span></p> <p>Using this result, we can simplify the matrix element as follows:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">⟨</mo><msubsup><mi>ψ</mi><mi>m</mi><mn>0</mn></msubsup><mi mathvariant="normal">∣</mi><mi>i</mi><mi>λ</mi><mo stretchy="false">[</mo><mi>A</mi><mo separator="true">,</mo><msub><mi>H</mi><mn>0</mn></msub><mo stretchy="false">]</mo><mi mathvariant="normal">∣</mi><msubsup><mi>ψ</mi><mi>n</mi><mn>0</mn></msubsup><mo stretchy="false">⟩</mo><mo>=</mo><mo>−</mo><mi>λ</mi><mi mathvariant="normal">ℏ</mi><mo stretchy="false">⟨</mo><msubsup><mi>ψ</mi><mi>m</mi><mn>0</mn></msubsup><mi mathvariant="normal">∣</mi><mi>A</mi><mfrac><mrow><mi mathvariant="normal">∂</mi><msubsup><mi>ψ</mi><mi>n</mi><mn>0</mn></msubsup></mrow><mrow><mi mathvariant="normal">∂</mi><mi>t</mi></mrow></mfrac><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle\psi_m^0|i\lambda[A,H_0]|\psi_n^0\rangle=-\lambda\hbar\langle\psi_m^0|A\frac{\partial\psi_n^0}{\partial t}\rangle </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord mathnormal">iλ</span><span class="mopen">[</span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">⟩</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord">−</span><span class="mord mathnormal">λ</span><span class="mord">ℏ</span><span class="mopen">⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⟩</span></span></span></span></span></p> <h3><strong>Evaluation of the Time Derivative</strong></h3> <p>To evaluate the time derivative <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><msubsup><mi>ψ</mi><mi>n</mi><mn>0</mn></msubsup></mrow><mrow><mi mathvariant="normal">∂</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\partial\psi_n^0}{\partial t}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4791em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1341em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight">t</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.5102em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.214em;margin-left:-0.0359em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>, we need to use the time-dependent Schrödinger equation:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>i</mi><mi mathvariant="normal">ℏ</mi><mfrac><mrow><mi mathvariant="normal">∂</mi><msubsup><mi>ψ</mi><mi>n</mi><mn>0</mn></msubsup></mrow><mrow><mi mathvariant="normal">∂</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mi>H</mi><mn>0</mn></msub><msubsup><mi>ψ</mi><mi>n</mi><mn>0</mn></msubsup></mrow><annotation encoding="application/x-tex">i\hbar\frac{\partial\psi_n^0}{\partial t}=H_0\psi_n^0 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord mathnormal">i</span><span class="mord">ℏ</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1111em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span></span></p> <p>Using this result, we can simplify the matrix element as follows:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">⟨</mo><msubsup><mi>ψ</mi><mi>m</mi><mn>0</mn></msubsup><mi mathvariant="normal">∣</mi><mi>i</mi><mi>λ</mi><mo stretchy="false">[</mo><mi>A</mi><mo separator="true">,</mo><msub><mi>H</mi><mn>0</mn></msub><mo stretchy="false">]</mo><mi mathvariant="normal">∣</mi><msubsup><mi>ψ</mi><mi>n</mi><mn>0</mn></msubsup><mo stretchy="false">⟩</mo><mo>=</mo><mi>λ</mi><msup><mi mathvariant="normal">ℏ</mi><mn>2</mn></msup><mo stretchy="false">⟨</mo><msubsup><mi>ψ</mi><mi>m</mi><mn>0</mn></msubsup><mi mathvariant="normal">∣</mi><mi>A</mi><mfrac><mrow><msub><mi>H</mi><mn>0</mn></msub><msubsup><mi>ψ</mi><mi>n</mi><mn>0</mn></msubsup></mrow><msubsup><mi>E</mi><mi>n</mi><mn>0</mn></msubsup></mfrac><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle\psi_m^0|i\lambda[A,H_0]|\psi_n^0\rangle=\lambda\hbar^2\langle\psi_m^0|A\frac{H_0\psi_n^0}{E_n^0}\rangle </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord mathnormal">iλ</span><span class="mopen">[</span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">⟩</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4241em;vertical-align:-0.933em;"></span><span class="mord mathnormal">λ</span><span class="mord"><span class="mord">ℏ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.453em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.933em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⟩</span></span></span></span></span></p> <h3><strong>Simplification of the Matrix Element</strong></h3> <p>Using the properties of the operators <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>H</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">H_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, we can simplify the matrix element as follows:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">⟨</mo><msubsup><mi>ψ</mi><mi>m</mi><mn>0</mn></msubsup><mi mathvariant="normal">∣</mi><mi>i</mi><mi>λ</mi><mo stretchy="false">[</mo><mi>A</mi><mo separator="true">,</mo><msub><mi>H</mi><mn>0</mn></msub><mo stretchy="false">]</mo><mi mathvariant="normal">∣</mi><msubsup><mi>ψ</mi><mi>n</mi><mn>0</mn></msubsup><mo stretchy="false">⟩</mo><mo>=</mo><mi>λ</mi><msup><mi mathvariant="normal">ℏ</mi><mn>2</mn></msup><mo stretchy="false">⟨</mo><msubsup><mi>ψ</mi><mi>m</mi><mn>0</mn></msubsup><mi mathvariant="normal">∣</mi><mi>A</mi><msub><mi>H</mi><mn>0</mn></msub><msubsup><mi>ψ</mi><mi>n</mi><mn>0</mn></msubsup><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle\psi_m^0|i\lambda[A,H_0]|\psi_n^0\rangle=\lambda\hbar^2\langle\psi_m^0|AH_0\psi_n^0\rangle </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord mathnormal">iλ</span><span class="mopen">[</span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">⟩</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord mathnormal">λ</span><span class="mord"><span class="mord">ℏ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">⟩</span></span></span></span></span></p> <h3><strong>Evaluation of the Matrix Element</strong></h3> <p>To evaluate the matrix element <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><msubsup><mi>ψ</mi><mi>m</mi><mn>0</mn></msubsup><mi mathvariant="normal">∣</mi><mi>A</mi><msub><mi>H</mi><mn>0</mn></msub><msubsup><mi>ψ</mi><mi>n</mi><mn>0</mn></msubsup><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle\psi_m^0|AH_0\psi_n^0\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">⟩</span></span></span></span>, we need to use the properties of the operators <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>H</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">H_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. Since <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> is an operator that represents a rotation, we can assume that it commutes with the unperturbed Hamiltonian <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>H</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">H_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. Therefore, we can write:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">⟨</mo><msubsup><mi>ψ</mi><mi>m</mi><mn>0</mn></msubsup><mi mathvariant="normal">∣</mi><mi>A</mi><msub><mi>H</mi><mn>0</mn></msub><msubsup><mi>ψ</mi><mi>n</mi><mn>0</mn></msubsup><mo stretchy="false">⟩</mo><mo>=</mo><mo stretchy="false">⟨</mo><msubsup><mi>ψ</mi><mi>m</mi><mn>0</mn></msubsup><mi mathvariant="normal">∣</mi><msub><mi>H</mi><mn>0</mn></msub><mi>A</mi><msubsup><mi>ψ</mi><mi>n</mi><mn>0</mn></msubsup><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle\psi_m^0|AH_0\psi_n^0\rangle=\langle\psi_m^0|H_0A\psi_n^0\rangle </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">⟩</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">⟩</span></span></span></span></span></p> <p>Using this result, we can simplify the matrix element as follows:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">⟨</mo><msubsup><mi>ψ</mi><mi>m</mi><mn>0</mn></msubsup><mi mathvariant="normal">∣</mi><mi>i</mi><mi>λ</mi><mo stretchy="false">[</mo><mi>A</mi><mo separator="true">,</mo><msub><mi>H</mi><mn>0</mn></msub><mo stretchy="false">]</mo><mi mathvariant="normal">∣</mi><msubsup><mi>ψ</mi><mi>n</mi><mn>0</mn></msubsup><mo stretchy="false">⟩</mo><mo>=</mo><mi>λ</mi><msup><mi mathvariant="normal">ℏ</mi><mn>2</mn></msup><mo stretchy="false">⟨</mo><msubsup><mi>ψ</mi><mi>m</mi><mn>0</mn></msubsup><mi mathvariant="normal">∣</mi><msub><mi>H</mi><mn>0</mn></msub><mi>A</mi><msubsup><mi>ψ</mi><mi>n</mi><mn>0</mn></msubsup><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle\psi_m^0|i\lambda[A,H_0]|\psi_n^0\rangle=\lambda\hbar^2\langle\psi_m^0|H_0A\psi_n^0\rangle </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord mathnormal">iλ</span><span class="mopen">[</span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">⟩</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord mathnormal">λ</span><span class="mord"><span class="mord">ℏ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">⟩</span></span></span></span></span></p>$$