Obtain the first-order energy correction of the first excited state of an isotropic two-dimensional harmonic oscillator subjected to a shape perturbation $V(x, y) = \lambda x^4$.
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Step 1: The first-order energy correction is given by: $E^{(1)} = \langle \psi^{(0)} | V | \psi^{(0)} \rangle$ where $\psi^{(0)}$ is the unperturbed wavefunction of the first excited state. Show more…
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The potential energy for the anharmonic oscillator is given by V(x) = kx^4. Solve for the first order correction to the ground state energy and the first excited state energy for the anharmonic oscillator using perturbation theory (from question 2d). Which has a bigger correction to the harmonic oscillator energy? It is recommended that you use Mathematica for the integrals in this question (include your code with the assignment if you do); although they can be done by hand with the use of integral tables and making use of the properties of odd/even functions.
Adi S.
Consider a particle in a two-dimensional potential \[ V_{0}=\left\{\begin{array}{ll} 0, & \text { for } 0 \leq x \leq L, 0 \leq y \leq L \\ \infty, & \text { otherwise } \end{array}\right. \] Write the energy eigenfunctions for the ground state and the first excited state. We now add a time-independent perturbation of the form \[ V_{1}=\left\{\begin{array}{ll} \lambda x y, & \text { for } 0 \leq x \leq L, 0 \leq y \leq L \\ 0, & \text { otherwise } \end{array}\right. \] Obtain the zeroth-order energy eigenfunctions and the first-order energy shifts for the ground state and the first excited state.
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