The sudden approximation

Problem:

Two infinite potential wells are extending from x = -a to x =  0 and from x = 0 to  x = a, respectively.  A particle is in its ground state in the left well.  At t = 0 the barrier at x = 0 is removed.  What is the probability of finding the particle in the first excited state of the new well?

Solution:

Problem:

Before t = 0, the ground state wave function of a particle with mass m is
ψ(x,t) = N(sin(πx/L) + sin(2πx/L))exp(-iEt/ħ) for 0 < x < L, ψ(x,t) = 0 everywhere else. 
Set the potential energy of the particle to zero at x = L/2.
(a)  Find the ground state energy E and the potential energy function U(x) before t = 0.
(b)  At t = 0 the potential energy function U(x) suddenly changes to U(x) = 0 for 0 < x < L, U(x) = ∞ everywhere else.  For t > 0, find the probability of measuring the different energy eigenvalues of the particle in this infinite well for t > 0.
(c)  Find P(x,t) for t < 0 and t > 0.

Solution:

Problem:

A particle of mass m is in the ground state of a harmonic oscillator with spring constant k = mω2.  
At t = 0, the spring constant changes suddenly to k' = 4k.  
Find the probability that the oscillator remains in its ground state.

-∞ exp(-x2/a2)dx = √(π) a

Solution:

Problem:

Consider a one-dimensional quantum particle with a Hamiltonian H = T + U(x),
T = -(ħ2/2m)(∂2/∂x2).  Suppose that m suddenly changes from m0 to m1 = m0/λ at t = 0.
Assuming the particle was in the ground state at t < 0, find
(i) the probability it remains in the ground state at t > 0 and
(ii) the change in the energy expectation value <H>.  Consider two cases:
(a)  The infinite well, U(x) = 0 for 0 < x < L, and U(x) infinite otherwise.
(b)  The parabolic well,  U(x) = Cx2/2.

Hint:  for any a we have  ∫-∞dx exp(-ax2) = (π/a)½.

Solution:

Problem:

The ground state wave function for a hydrogen like atom is
Φ100(r) = (1/√π)(Z/a0)3/2 exp(-Zr/a0),
where  a0 = ħ2/(μe2) and μ is the reduced mass, μ ~ me = mass of the electron. 
(a)  What is the ground state wave function of tritium?
(b)  What is the ground state wave function of 3He+?
(c)  An electron is in the ground state of tritium.  A nuclear reaction instantaneously changes the nucleus to 3He+.  Assume the beta particle and the neutrino are immediately removed from the system.  Calculate the probability that the electron remains in the ground state of 3He+.

Solution: