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:

• Concepts:
  Postulates of quantum mechanics
• Reasoning:
  We use the given wave function to calculate the mean value and rms deviation of observables.
• Details of the calculation:
  (a)  We need  A²∫_-a^a(a² − x²)²dx = 2∫₀^a (a² − x²)²dx = A²16a⁵/15 = 1.
  A = √15 a^-5/2/4.
  (b)  <x> = ∫_-∞^∞|Ψ(x)|² x dx = A²∫_-a^a(a² - x²)² x dx = 0.
  |Ψ|² is even, x is odd and the integral is zero.
  (c)  <p> = (ħ/i)∫_-∞^∞Ψ*(x) (dΨ(x)/dx) dx = -A²(ħ/i)∫_-a^a2x (a² - x²) dx = 0.
  (d)  <x²> =  ∫_-∞^∞|Ψ(x)|² x² dx = A²∫_-a^a(a² - x²)² x² dx = a²/7.
  (e)  <p²> = -ħ²∫_-∞^∞Ψ*(x) (d²Ψ(x)/dx²) dx = A²ħ²∫_-a^a2(a² - x²) dx = (5/2)(ħ²/a²).
  (f)  Δx = (<x²> - <x>²)^½ = a/√7.
  (g)  Δp = (<p²> - <p>²)^½ = (5/2)^½(ħ/a).
  (h)  The uncertainty principle is satisfied.
  ΔxΔp = (a/√7)(5/2)^½(ħ/a) = (5/14)^½ħ > ħ/2.

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:

• Concepts:
  The postulates of Quantum Mechanics
• Reasoning:
  When a physical quantity described by the operator A is measured on a system in a normalized state |ψ>, the probability of measuring the eigenvalue a_n is given by
  P(a_n) = Σ_i=0^gn|<u_nⁱ|ψ>|²,  where {|u_nⁱ>} (i=1,2,...,g_n) is an orthonormal basis in the eigensubspace E_n associated with the eigenvalue a_n.
  If a measurement on a system in the state |ψ> gives the result a_n, then the state of the system immediately after the measurement is the normalized projection of |ψ> onto the eigensubspace associated with a_n.
• Details of the calculation:
  We assume the system has no time to evolve between measurements.
  (a)  After measuring a₁, the system is in the eigenstate ψ₁.
  (b)  If B is now measured, the probability of obtaining b₁ is 9/25 and the probability of obtaining b₂ is 16/25.
  (c)  There are two path.  If b₁ is measured the system is in the state φ₁ = (3ψ₁ + 4ψ₂)/5.
  The probability of measuring a₁ after measuring b₁ is 9/25.
  If b₂ is measured the system is in the state φ₂ = (4ψ₁ - 3ψ₂)/5.
  The probability of measuring a₁ after measuring b₂ is 16/25.
  The probability of getting a₁ again after a measurement of B is (9/25)² + (16/25)² = 0.5392.

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 =  

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 =  

,

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

Solution:

• Concepts:
  Mathematical foundations of quantum mechanics
• Reasoning:
  This is a linear algebra problem.
• Details of the calculation:
  (a)  To find the eigenvalues of an n×n matrix A requires solving the characteristic (also termed secular) equation det(A - λI) = 0 for λ, with I the n×n unit matrix,  For the general 2×2 matrix
  
  

  A =  

  	a	b
  	c	d

   

  we have
  

  	a - λ	b
  	c	d - λ

    = 0.

  
  
  (a - λ)(d - λ) - bc = 0.
  λ_± = ½(a + d) ± ((a + d)² - 4(ad - bc))^½ = ½T ± (¼T² - D)^½,
  where T = Tr(A) = a + b and D = det(A) = ad - bc.
  Applying this to the matrix

  
  

  A =  

  	0	m
  	m	M

  
  
  and assuming M >> m gives T = M,  D = -m²,
  λ_± = ½M ± (¼M² + m²) = ½M ± ½M(1 + 4m²/M²)^½ ≈ ½M ± ½M(1 + 2m²/M²)
  = ½M ± (½M + m²/M).
  The eigenvalues therefore are λ₊ ≈ M,  λ_- ≈ m²/M,  |λ₊| >> |λ_-|.
  
  (b)  Parameterize the matrix in terms of complex parameters a, b, c, and d, and require the unitarity condition UU^† = I.

  U =  

  	a	b
  	c	d

        U^† =  

  	a*	c*
  	b*	d*

  .

   

  UU^† = I --> aa^* + bb^* = 1, cc^* + dd^* =  1, ac^* + bd^* = 0.  b = -c^*a/d^*.
  det(U) = 1 --> ad - bc = 1,  ad + |c|²a/d^* = 1,  a(|d|² + |c|²) = d^*,  a = d^*, b = -c^*.
  Therefore

  U =  

  	a	b
  	-b*	a*

  
   

  |a|² + |b|² = 1.

Problem 4:

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

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

Solution:

• Concepts:
  Functions of operators, change of basis
• Reasoning:
  Use the eigenbasis of H to evaluate a function of the operator H.
• Details of the calculation:
  The eigenvalues of H are E, where
   

  	-E	2	0
  	2	-E	0
  	0	0	1-E

    = 0.

  
  

  
  E²(1 - E) - 4(1 - E) = 0.  The solutions to this equation are E₁ = -2, E₂ = 1, E₃ = 2.
  The corresponding normalized eigenvectors are

  |E₁> = 2^-½  

  	-1
  	1
  	0

  ,   |E₂> =  

  	0
  	0
  	1

  ,   |E₃> = 2^-½  

  	1
  	1
  	0

  .

  
  

  The matrix of H in its eigenbasis, with the eigenvectors ordered as {|E₁>, |E₂>, |E₃>}, is

  H =  

  	-2	0	0
  	0	1	0
  	0	0	2

  .  

  
  

  The matrix of the evolution operator in the eigenbasis of H is

  U' = exp(-iHt) =  

  	exp(i2t)	0	0
  	0	exp(-it)	0
  	0	0	exp(-i2t)

  .  

  
  

  The matrix of the evolution operator in the original {|1>, |2>, |3>} basis is
  U= DU'D^†,  where D is the unitary transformation from the {|1>, |2>, |3>} basis to the {|E₁>, |E₂>, |E₃>} basis.
  The matrix of D is

  D  =  

  	-2-½	0	2-½
  	2-½	0	2-½
  	0	1	0

  .  

  
  

  The matrix of D^† is

  D^† =  

  	-2-½	2-½	0
  	0	0	1
  	2-½	2-½	0

  .  

  
  

  Therefore the matrix of the evolution operator in the original  {|1>, |2>, |3>} basis is

  U =  

  	cos(2t)	-isin(2t)	0
  	-isin(2t)	cos(2t)	0
  	0	0	exp(-it)

  .  

  
  

   

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 = ħωa^†a, where the operators a and a^†are 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:

• Concepts:
  Fundamental assumptions of QM
• Reasoning:
  We are given enough information to construct the matrix of the Hermitian operator H in the given basis. 
• Details of the calculation:
  (a)  H|0> = ħωa^†a|0> = 0  -->  E₀ = 0,  |0> is an eigenvector of H.
  H|1> = ħωa^†a|1> = ħω|1>  -->  E₁ = ħω,  |1> is an eigenvector of H.
  (b)  For any operator O we have = ∑_n ∑_m|n><n|O|m><m|.
  H = 0|0><0| + ħω|1><1| = ħω|1><1|.
  a = |0><1|,  a^† = |1><0|.
  (c)  aa^† = |0><0|,  a^†a = |1><1|,  {a, a^†} = aa^† + a^†a = I.