3. Perturbation Theory. Consider a quantum mechanical particle in a square po- tential well i.e. the...

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3. Perturbation Theory. Consider a quantum mechanical particle in a square po- tential well i.e. the 2D 'particle-in-a-box' of length 2a, with Vo(x, y) = 0 for a < x, y < a and V(x, y) = otherwise. (a) Write the total wavefunction of the system. You can write it as the product of two orthogonal Eigenfunctions derived in class. (b) We now add a point-like perturbation at the position (1/2a, 1/2a) with V(x, y) Vp8(x-a)(ya), where Vp is an arbitrary potential and (...) denotes the delta-function. Use perturbation theory to calculate the energy-correction to the ground state up to first order. The following property of the delta-function will use useful: | f(x)(x a)dx = (a) (1) (c) Bonus: Read up on perturbation theory for degenerate states (Atkins 6.4) and calculate the energy correction for the first excited states with quantum numbers nx = 1, ny = 2 and n = 2, ny = 1. = 3. Perturbation Theory. Consider a quantum mechanical particle in a square po- tential well i.e. the 2D 'particle-in-a-box' of length 2a, with Vo(x, y) = 0 for a < x, y < a and V(x, y) = otherwise. (a) Write the total wavefunction of the system. You can write it as the product of two orthogonal Eigenfunctions derived in class. (b) We now add a point-like perturbation at the position (1/2a, 1/2a) with V(x, y) Vp8(x-a)(ya), where Vp is an arbitrary potential and (...) denotes the delta-function. Use perturbation theory to calculate the energy-correction to the ground state up to first order. The following property of the delta-function will use useful: | f(x)(x a)dx = (a) (1) (c) Bonus: Read up on perturbation theory for degenerate states (Atkins 6.4) and calculate the energy correction for the first excited states with quantum numbers nx = 1, ny = 2 and n = 2, ny = 1. =