Using the time independent non-degenerate perturbation theory to deduce the 2nd order correction and then find the 1st and 2nd order correction to wave function for a particle in s... Using the time independent non-degenerate perturbation theory to deduce the 2nd order correction and then find the 1st and 2nd order correction to wave function for a particle in slanted potential box.
Understand the Problem
The question is asking to apply time-independent non-degenerate perturbation theory to deduce the second order correction and then find the first and second order corrections to the wave function for a particle in a slanted potential box.
Answer
The first and second order corrections for the wave function can be expressed as: 1st order: $$| \psi_n^{(1)} \rangle = \sum_{k \neq n} \frac{\langle k | V | n \rangle}{E_n^{(0)} - E_k^{(0)}} | k \rangle $$ 2nd order: $$| \psi_n^{(2)} \rangle = | \psi_n^{(1)} \rangle + \text{corrections from } | \psi_n^{(1)} \rangle $$
Answer for screen readers
The first and second order corrections to the wave function for a particle in a slanted potential box depend on the calculations of the perturbation terms as discussed.
-
First order correction: $$ E_n^{(1)} = \langle n | V | n \rangle $$
-
Second order correction: $$ E_n^{(2)} = \sum_{k \neq n} \frac{|\langle k | V | n \rangle|^2}{E_n^{(0)} - E_k^{(0)}} $$
The wavefunction corrections are given by:
-
First order: $$ | \psi_n^{(1)} \rangle = \sum_{k \neq n} \frac{\langle k | V | n \rangle}{E_n^{(0)} - E_k^{(0)}} | k \rangle $$
-
Second order: $$ | \psi_n^{(2)} \rangle = | \psi_n^{(1)} \rangle + \sum_{k \neq n} \frac{\langle k | V | n \rangle}{E_n^{(0)} - E_k^{(0)}} | k \rangle $$
Steps to Solve
-
Understanding Perturbation Theory In quantum mechanics, perturbation theory is used to approximate the behavior of a system when it is subject to a small perturbation. The time-independent non-degenerate perturbation theory will help us determine the corrections to the energy levels and wave functions.
-
Identify the Unperturbed Hamiltonian The unperturbed Hamiltonian for a particle in a box of width (L) can be expressed as: $$ H_0 = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} $$ The corresponding eigenfunctions and eigenvalues are known.
-
Define the Perturbation For a slanted potential box, we define the perturbation ( V(x) ) as: $$ V(x) = m \cdot g \cdot x $$ where ( g ) is the gravitational constant or the slope of the potential.
-
First Order Energy Correction The first order energy correction ( E_n^{(1)} ) is given by: $$ E_n^{(1)} = \langle n | V | n \rangle $$ Calculate this using the unperturbed wave functions.
-
Second Order Energy Correction The second order energy correction ( E_n^{(2)} ) is calculated as: $$ E_n^{(2)} = \sum_{k \neq n} \frac{|\langle k | V | n \rangle|^2}{E_n^{(0)} - E_k^{(0)}} $$ where ( |k\rangle ) are the other eigenstates.
-
First Order Wave Function Correction The first order correction to the wavefunction ( | \psi_n^{(1)} \rangle ) is given by: $$ | \psi_n^{(1)} \rangle = \sum_{k \neq n} \frac{\langle k | V | n \rangle}{E_n^{(0)} - E_k^{(0)}} | k \rangle $$
-
Second Order Wave Function Correction The second order correction to the wave function ( | \psi_n^{(2)} \rangle ) includes contributions from the first order term: $$ | \psi_n^{(2)} \rangle = \sum_{k \neq n} \frac{\langle k | V | n \rangle}{E_n^{(0)} - E_k^{(0)}} | k \rangle + \text{corrections from } \psi_n^{(1)} $$
The first and second order corrections to the wave function for a particle in a slanted potential box depend on the calculations of the perturbation terms as discussed.
-
First order correction: $$ E_n^{(1)} = \langle n | V | n \rangle $$
-
Second order correction: $$ E_n^{(2)} = \sum_{k \neq n} \frac{|\langle k | V | n \rangle|^2}{E_n^{(0)} - E_k^{(0)}} $$
The wavefunction corrections are given by:
-
First order: $$ | \psi_n^{(1)} \rangle = \sum_{k \neq n} \frac{\langle k | V | n \rangle}{E_n^{(0)} - E_k^{(0)}} | k \rangle $$
-
Second order: $$ | \psi_n^{(2)} \rangle = | \psi_n^{(1)} \rangle + \sum_{k \neq n} \frac{\langle k | V | n \rangle}{E_n^{(0)} - E_k^{(0)}} | k \rangle $$
More Information
Perturbation theory is indispensable in quantum mechanics as it allows the analysis of complex systems by approximating effects of small disturbances. The slanted potential introduces linear variations in potential, leading to shifts in energy levels and changes in wavefunctions.
Tips
- Miscalculating Matrix Elements: Be careful to accurately compute the integrals for ( \langle k | V | n \rangle ).
- Ignoring Non-degenerate Conditions: Ensure that the non-degenerate assumption holds, as it simplifies calculations. If not, the degenerate perturbation theory methods should be used instead.
AI-generated content may contain errors. Please verify critical information
