Mixed method problems

Problem:

A one-dimensional quantum oscillator consisting of a mass m suspended from a spring with spring constant k is initially in its lowest energy state.  At t = 0, the upper end of the spring is suddenly raised a distance d during a time interval which is very short compared to an oscillator period.
(a)  Give explicit expressions for the time-dependent eigenstates of the Hamiltonian for t < 0 and those for t > 0 and discuss the relationship between them in the context of this problem.
(b) Write down an integral for the probability that a transition has occurred to the first excited state as a result of the disturbance at t = 0.
(c)  Now consider a situation involving the above disturbance at t = 0, followed by an interval from 0 < t < T, where T is large compared with a ground-state period and during which the upper end of the spring remains fixed at the disturbed or stretched position.  At t = T the upper end of the spring is suddenly returned to its lower, original position.
Derive the probability amplitude for the oscillator to be found in its first exited state at times t > T.

The stationary states of H0 = p2/(2m) + kx2/2 are un(x) = NnHn(αx)exp(-α2x2/2)
with Nn = [α/(π½2nn!)]½, α = (mk/ħ2)¼  and Hn(αx) a Hermite polynomial.
Recursion relation: Hn+1(αx) = 2αxHn(αx) - 2nHn-1(αx).
H0(αx) = 1,   H1(αx) = 2αx,   H2(αx) = 4(αx)2 - 2.

Solution:

Problem:

(a)  An electron is located in the ground state in a 1D potential well,
U(x) = 0, 0 < x < L, U(x) = ∞ elsewhere. 
Instantaneously the well becomes 1.5 times wider.  What is the probability for the electron to go directly to ground state in this new well?

(b)  An electron is located in the ground state in a 1D potential well,
U(x) = 0, 0 < x < L, U(x) = ∞ elsewhere.   
For a time interval ∆t, the well bottom is disturbed, and the potential energy function becomes U(x) = 0, 0 < x < L/2, U(x) = U0, L/2 < x < L, with U0 << than the ground state energy.  After the time interval ∆t the disturbance is removed.  What is the probability for the electron to be in the first excited state of the well at a time t = 2∆t? 

Solution:

Problem:

Consider a system of two non-identical spin ½ particles.  For t < 0 the particles do not interact and the Hamiltonian may be taken to be zero.  For t > 0 the Hamiltonian is given by

H = (4Δ/ħ2) S1S2,

where Δ is a constant.  For t < 0 the state of the system is |+ –>. 
(a)  For t > 0, find, as a function of time, the probability for finding the system in each of the states |+ +>, |+ –>, |– +>, and |– –>, by solving the problem exactly.
(b)  For t > 0, find, as a function of time, the probability for finding the system in each of the states |+ +>, |+ –>, |– +>, and |– –>, using first-order time dependent perturbation theory with H a perturbation that is switched on at t = 0.
(c)  Under what condition is the perturbation calculation a bad approximation to the exact solution, and why?

Solution:

Problem:

Consider a 3-state system with a Hamiltonian

image

in the {|i>} orthonormal basis, i = 1, 2, 3, with B, A > 0.  For < 0, the system is in state |1>.
At t = 0 the Hamiltonian suddenly changes to

image

with C << A, B, C > 0.
At t = T the Hamiltonian changes back to H0.
(a)  What are the eigenstates and eigenvalues if H expressed in terms of the {|i>} basis vectors?
(b)  For each of the eigenstates of H, what are the probabilities that the system will be found in this state for 0 < t < T?
(c)  Using first order time-dependent perturbation theory, what is the probability of finding the system in state |3> at t’ > T?
(d)  Solve part (c) exactly.

Solution:

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