Problem 1: 

Consider a spin coupled to a magnetic field via the Hamiltonian H = γS∙B, where the three-dimensional magnetic field is
B = (B_x, B_y, B_z) = B₀(1, 0, 0),
B₀ is a constant, and γ is another constant. 
The three-dimensional spin vector is S = (S_x, S_y, S_z), and the S_x, S_y, S_z are
the usual 2 x 2 spin matrices.
(a)  In the eigenbasis of S_z, find the matrix of H and its eigenvalues.
(b)  In the eigenbasis of S_z, find the two normalized eigenvectors of the H.

Solution:

• Concepts:
  The two-dimensional state space of a spin ½ particle, the postulates of quantum mechanics
• Reasoning:
  H = γS∙B = γB₀S_x.
  The eigenstates of H are the eigenstates of S_x. 
  The  eigenvalues of any S_x are ±ħ/2.
• Details of the calculation:
  (a)  The eigenvalues of H are = ±γB₀ħ/2.
  The matrix of H is the matrix of γB₀S_x or the matrix of (γB₀ħ/2)σ_x.
  The matrix of the Pauli matrix σ_x is
  
  H and σ_x have the same eigenvectors.
  (b)  The normalized eigenvectors of σ_x are |±>_x = (1/√2)(|+> ± |->).

Problem 2:

Suppose an electron is in a state described by the wave function

ψ = (1/√4π)(e^iφsinθ + cosθ)g(r),

where  ∫₀^∞ |g(r)|²r²dr = 1, and φ, θ are the azimuth and polar angles respectively.

(a)  What are the possible results of a measurement of the z-component L_z of the angular momentum of the electron in this state?
(b)  What is the probability of obtaining each of the possible results in part (a)?
(c)  What is the expectation value of L_z ?

Solution:

• Concepts:
  The spherical harmonics, the postulates of quantum mechanics
• Reasoning:
  If we want to find the probability of measuring the eigenvalue a of an observable A we must express the wave function of the system as a linear combination of eigenfunctions of A.  Then P(a) = Σ_i|<aⁱ|ψ>|².
  Express the wave function in terms of spherical harmonics.
  Y_1±1 = ∓(3/8π)^½sinθ exp(±iφ),  Y₁₀ = (3/4π)^½cosθ.
• Details of the calculation:
  (a)    ψ = (1/3)^½(-√2 Y₁₁ + Y₁₀)g(r),  the possible results for a measurement of L_z are 0 and ħ.
  (b)    The wavefunction ψ is normalized.  The probability of measuring 0 is 1/3, the probability of measuring ħ is 2/3.
  (c)     The expectation value <L_z> = 0*P(0) + ħP(ħ) = (2/3)ħ.

Problem 3:

A particle of mass m is confined to an infinite one-dimensional potential well of width L, i.e. U(x) = 0,  0 < x < L,  U(x) = ∞ everywhere else.
It is in the ground state of this well.
Find the uncertainty (rms deviation) ∆p in the momentum of this particle.

Solution:

• Concepts:
  Fundamental assumptions of QM, the eigenstates of the 1D infinite well
• Reasoning:
  The eigenfunctions of the infinite well are
  ψ_n(x) = (2/L)^½sin(nπx/L), E_n = n²π²ħ²/(2mL²).
  For any operator A we have ∆A = (<(A - <A>)²>)^½ = (<A²> - <A>²)^½.
• Details of the calculation:
  ∆p =  (<p²> - <p>²)^½.
  <p> = 0 for any stationary state.
  <p²> =  -(2ħ²/L)∫₀^L dx  sin(πx/L) (∂²/∂x²) sin(πx/L)
  = (2ħ²π/L²)∫₀^π dy  sin²(y) = (ħ²π²/L²).
  ∆p = (ħπ/L).

Problem 4:

Assume that the highest energy (Fermi energy) of an electron inside a block of metal is 5 eV and that the work function of the metal, i.e. the additional energy that is necessary to remove an electron from the metal, is 3 eV.
(a)  Estimate the distance through which the wave function of an electron at the Fermi level penetrates the barrier responsible for the work function, assuming that the width of the 3 eV barrier is much greater than the penetration distance.  (In this estimate you can assume that the barrier is a simple step function.)
(b)  Estimate the transmission coefficient (i.e., provide a numerical value for it) for an electron at the Fermi level if the 3 eV barrier is now assumed have a width of 20 Å.
Note: the probability of transmission through a barrier of width a is given by
T = 4E(U₀ - E) / [(U₀²sinh²[(2m(U₀ - E)/ħ²)^½ a] + 4E(U₀ - E)],
or
T = (4E/U₀)(1 - E/U₀) / [sinh²[(2m(U₀ - E)/ħ²)^½ a] + (4E/U₀)(1 - E/U₀)].

Solution:

• Concepts:
  This is a "square potential" problem.
• Reasoning:
  We are given a piecewise constant potential.
• Details of the calculation:
  Let region 1 be the region to the left of the step and region 2 the region to the right of the step.  For an electron at the Fermi level with E < U₀ we define
  k₁² = (2m/ħ²)E and ρ₂² = (2m/ħ²)(U - E).
  Then  
  Φ₁(x) = A₁exp(ik₁x) + A₁'exp(-ik₁x) 
  and  
  Φ₂(x) = B₂exp(ρ₂x) + B₂'exp(-ρ₂x)  
  are the most general solutions.  We need B₂ = 0 for the solution to remain finite at x --> ∞. 
  Φ₂(x) = B₂'exp(-ρ₂x).  The penetration depth is on the order of 1/ρ₂.
  ρ₂ = ((2m(3 eV)/ħ²)^½ = (2*9.1*10^-31 kg*3*1.6*10^-19J/(1.054*10^-34 Js)²)^½
  = 8.87*10⁹/m.
  1/ρ₂ = 1.13*10^-10 m = 1.13 Å.
  (b)  For E < U₀ the probability of transmission through a barrier of width a is
  T = (4E/U₀ )(1 - E/U₀) / [sinh²[(2m(U₀ - E)/ħ²)^½ a] + (4E/U₀)(1 - E/U₀)]. 
  Here U₀ = 8 eV and E = 5 eV and a = 2*10^-9 m.
  T = (20/8)(1 - 5/8 ) / [sinh²[ρ₂*2*10^-9] + (20/8)(1 - 5/8)] = 1.4*10^-15.
  
  Since ρ₂a = 17.74 we can approximate sinh(ρ₂a) = ½(exp(ρ₂a) - exp(-ρ₂a)) by ½exp(ρ₂a) and the expression for T by
  T = (16E/U₀)(1 - E/U₀)exp(-2((2m/ħ²)(U₀ - E))^½a) = 1.4*10^-15.
  
  Aside:
  The tunneling probability strongly depends on the barrier's width and height.  Most particles approaching the barrier are reflected, but a few tunnel through the barrier.  Often we estimate the probability P of finding the particle with energy E on the other side of the energy barrier of height U > E as P = exp(-2αd), where d is the width of the barrier and α = [(2m/ħ² )(U - E)]^½.  That is a reasonable order of magnitude estimate if E is just slightly smaller than U and (16E/U₀)(1 - E/U₀) is approximately equal to 1.