Assignment 5

Problem 1:

A particle is represented (at time t = 0) by the wave function

ψ(x,t) = A(a2 - x2) if -a < x < a, ψ(x,t) = 0 otherwise.

(a)  Determine the normalization constant A.
(b)  What is the expectation value of x (at time t = 0)?
(c)  What is the expectation value of p (at time t = 0)?
(d)  Find the expectation value of x2.
(e)  Find the expectation value of p2.
(f)   Find the uncertainty in x (Δx).
(g)  Find the uncertainty in p (Δp).
(h)  Check that your results are consistent with the uncertainty principle.

Solution:

Problem 2:

An operator A, representing observable A, has two normalized eigenstates Ψ1 and Ψ2, with eigenvalues a1 and a2, respectively.  An operator B, representing observable B, has two normalized eigenstates Φ1 and Φ2, with eigenvalues b1 and b2, respectively.  The eigenstates are related by
Ψ1 = (3Φ1 + 4Φ2)/5,
Ψ2 = (4Φ1 - 3Φ2)/5.
(a)  Observable A is measured, and the value a1 is obtained.  What is the state of the system immediately after this measurement?
(b)  If B is now measured immediately following the A measurement, what are the possible results, and what is the probability of measuring each result?
(c)  Right after the measurement of B, A is measured again. What is the probability of getting a1?

Solution:

Problem 3:

(a)  Show that the eigenvalues of a general 2×2 matrix A can be expressed as
λ±= ½T ± (¼T2 - D)½,
where D is the determinant of A and T is the trace of A (sum of diagonal elements).

Show that the matrix

A =  

  0    m  
  m  M  

with M >> m has two eigenvalues, with one much larger than the other.

(b)  Show that the most general form of a 2×2 unitary matrix U with unit determinant can be parameterized as

U =  

   b  
  -b*  a*  

,

subject to the constraint aa* + bb* = 1, where * denotes complex conjugation. 

Solution:

Problem 4:

Consider a function F(z) which can be expanded in a power series in z, F(z) = ∑nfnzn.
The corresponding function of an operator Ω is defined as F(Ω) = ∑nfnΩn
Let |Φω> be an eigenvector of Ω with eigenvalue ω.  Then
F(Ω)|Φω> = ∑nfnΩnω> = ∑nfnωnω> = F(ω)|Φω> = number*|Φω>.
ω> is also an eigenvector of F(Ω).

Consider a three-state system.  In some orthonormal basis, {|1>, |2>, |3>}, the matrix of the Hamiltonian operator H is

H =  

  0    2    0   
  2    0  0   
  0    0    1   

.  

 

 

Choose units such that ħ = 1.   Find the matrix of the evolution operator in the same basis.

Solution:

Problem 5:

Consider a two state system with kets |0> and |1> defining a set which is assumed to be orthonormal and complete.
The Hamiltonian of the system is H = ħωaa, where the operators a and aare defined by
a|0> = |1>,  a|1> = |0>,  a|1> = a|0> = 0.
(a)  Find the eigenvalues of H.
(b)  Write down the Dirac (bra - ket) representation of the operators H, a, and a.
(c)  What are the anti-commutators of the operators a and a?

Solution: