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Q5: Consider a particle of mass m in a two-dimensional box having side length L and L₁ with L = 2L, and V=0 in the box, ∞ outside;
Suppose V=10 J in the box. What effect has this on the eigenvalues?
the eigen functions?

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Q1) Consider a particle in a one-dimensional infinite well with walls at x=0 and x=a. Suppose a perturbation is applied so that the potential energy is shifted by an amount (0) = n²h?/(2ma²) is the ground state energy (0) a H, = 10-3E," (x/a)², where E") of the unperturbed box. a) Calculate the first order correction to all excited state energies due to the perturbation. b) Calculate the second order correction to all excited state energies due to the perturbation. (Simplify your result as much as you can, f.eg. taking sin[(k ±n)T] = 0) c) Calculate the first order correction to the wave function due to the perturbation. d) Write down the approximate wave function of the perturbed well for the state with n = 3 corrected up to the 1.st order in perturbation theory . Consider only the first three non-zero terms in the sum.
2. Consider a particle of mass m in an infinite square well, 0(≤ x ≤a). At the time t = 0 the particle is in the ground (n = 1). Then at t> 0 a weak time-dependent external potential is turned on: H' = Axe T