Calculate the 1st order correction in energy of the 1st

Question

1. Consider a harmonic oscillator of frequency ω. So, the unperturbed Hamiltonian reads

H0 = Pˆ² /2m + 1/2 mω2Xˆ²             (1)

Now consider a perturbation of the form H1 = αXˆ³ + βXˆ⁴ where α, β are small parameters. Find out the 2nd order correction to the nth energy level. Discuss the validity of the perturbation series in this problem.

2. Consider a Hydrogen atom whose unperturbed Hamiltonian is given by

H₀ = p²/2µ-e²/(4πε0)r

where µ =mMp/m+Mp

Now consider the perturbation due to gravitational force H₁= -GmMp/r.

Calculate the 1st order correction in the nth energy level. Compare the result with the exact energy of nth level of the total Hamiltonian(H = H0 + H1 ).

3. Consider a 2d harmonic oscillator of frequency ω with the perturbation term

H₁ =1/2mω2(Xˆ ² - Yˆ ²) where  << 1. So, the full Hamiltonian reads

H = Pˆx² + Pˆy²/2m+ 1/2mω²(Xˆ² + Yˆ²) + 1/2mω²(Xˆ ² - Y^{ˆ 2})

Calculate the 1st order correction in energy of the 1st excited state of 2d harmonic oscillator which is doubly degenerate. Compare the result with the exact energy expression of H in 1st excited state. Also calculate the correction in the wavefunction of 1st excited state in the 1st order.

4. A particle of mass ‘m' is in the ground state of the 1D infinite square well potential of length ‘L'.

V(x) = {0   0≤x≤L

      = { ∞  otherwise

At t=0 the barrier, located at x=L is instantaneously pulled to x=2L so that the length becomes 2L.

(a) What is the probability that the particle will make a transition to the nth eigenstate of the new Hamiltonian.

(b) Find out the eigenstate which has the maximum transition probability. Calculate the average energy of the particle right after the change.2.

5. Consider a quantum system having two non-degenerate eigenstates a and b; their unperturbed energies being Ea⁽⁰⁾ and Eb⁽⁰⁾ (> E(0)a ) and corresponding eigenfunctions |ψ_a⁽⁰⁾> and |ψ_b⁽⁰⁾> respectively. Let's suppose that initially the system is in the ground state and at t=0 a constant perturbation H' is switched on.

(a) Calculate the probability of finding the system in the ground state and calculate the probability of the transition a → b at time t>0.

(b) Calculate the same(as in (a)) using first-order perturbative approach and verify with the exact results in (a) to the first order in H0 and for small time ‘t'.

Comment on the validity of the perturbation expansion in this case.

6. Consider a 2D harmonic oscillator of frequency ω; the Hamiltonian of the system reads

H₀ =Pˆx² + Pˆy² /2m+1/2mω2(Xˆ ² + Yˆ ²)

Let's suppose that initially the system is in the ground state of H0 and at t=0 the perturbation H₁(t) = 1/2mω²(t)(Xˆ ² - Yˆ ²) is switched on where (t) << 1.

(a) If ε(t) = ε0 i.e. H₁ is time-independent, calculate the probability that the system will make a transition to (i) the 1st excited state and (ii) the 2nd excited state(take the angular momentum carrying state) at time t>0.

(b) Calculate the same(as in (a)) when H₁ is time-dependent; time dependence in ε is given by, ε(t) = ε0(1 - e^-t/τ ) Also calculate the two limiting cases : (i) τ → 0(sudden quench) and (ii) τ → ∞(adiabatic limit).

Note : Use the common eigenbasis of H0 and Lz in your calculation.

7. Consider a 1D harmonic oscillator of frequency ω(t) being time-dependent; it is changing from ω0 to ω1(< ω0). So the Hamiltonian of the system reads : H =p²/2m +1/2mω²(t)x². Let's consider the ansatz ψ(x, t) = (Re(α(t))/πt²)^1/4 e^-α(t)x2/2l2e^iθ(t)

where α ∈ C, θ ∈ [0, 2π] and l = √h/mω0.

(a) Starting from the time evolution of ψ(x, t) under the Hamiltonian H, governed by the Schrödinger equation write down the equation of motion for Re(α(t)), Im(α(t)) and θ(t).(HINT: Collect the coefficients of the similar powers of x from both sides of the equation.)

(b) Substitute Re(α(t)) by 1/b2 (t) and find out the time evolution of b(t) in the case when the frequency is abruptly changed from ω0 to ω1 where ω1 << ω0 and t << 1/ω1 so that the expansion of the wave-function is practically free.