The Schroedinger equation

Problem:

(a)  Define what is meant by the term "stationary state" in quantum mechanics, and explain the distinction between the time-dependent and time-independent Schroedinger equation.
(b)  At time t = 0, the wave function of a particle in one dimension is ψ(x) = (u₁(x) + u₂(x))/√2, where u₁(x) and  u₂(x) are two solutions of the time-independent Schroedinger equation.  
For this particle, how does the probability density change with time?

Problem:

In relativistic mechanics, energy and momentum of a free particle are related by the expression E² = p²c² + m²c⁴.  Construct the relativistic analogue of the Schroedinger equation by introducing the appropriate operators.

Solution:

• Concepts:
  The momentum and energy operators, postulates of Quantum Mechanics
• Reasoning:
  How do we find the operator corresponding to a physical quantity that is classically defined?
  (a)  Express the physical quantity in terms of the fundamental dynamical variables r and the conjugate momenta p.
  (b)  Symmetrize the expression with respect to r and p, then replace the variables r and p with the operators R and P.
  Here r does not appear and no symmetrization is necessary.
• Details of the calculation:
  (p²c² + m²c⁴)ψ = E²ψ
  The momentum operator is (ħ/i)∇.  The energy operator is iħ∂/∂t.
  We therefore have
  (-c²ħ²∇² + m²c⁴)ψ = -(ħ²∂²/∂t²)ψ, or
  (∇² - m²c²/ħ²)ψ = (1/c²)(∂²/∂t²)ψ.

Problem:

Solution:

• Concepts:
  The Schroedinger equation, ρ(r,t) = |ψ(r,t)|².
• Reasoning:
  Using the Schroedinger equation we can find an expression for ∂ρ(r,t)/∂t in terms of space derivatives.
• Details of the calculation:
  (a)  The probability of finding a particle described by the normalized wave function ψ(r,t) in a volume d³r about r is given by dP(r,t) = |ψ(r,t)|²d³r.  The total probability of finding the particle at time t anywhere in space is ∫_{all space} |ψ(r,t)|²d³r = <ψ|ψ> = 1.
  Conservation of probability implies d<ψ|ψ>/dt = 0.
  (b)  Local conservation of a classical quantity is usually expressed through the equation ∇∙j = -(∂/∂t)ρ.  Here ρ(r,t) is the volume density and j(r,t) is the current density.
  Assuming local conservation of probability we write
  ∂ρ/∂t = ∂ψ*ψ/∂t = ψ*∂ψ/∂t + ψ∂ψ*/∂t
  = (1/(iħ))[ψ*((-ħ²/(2m))∇²ψ + Vψ) - ((-ħ²/(2m))∇²ψ + Vψ)*ψ]
  = (-ħ/(2im)[ψ*∇²ψ -(∇²ψ)*ψ], since V is real.
  ∂ρ/∂t = (-ħ/(2im)∇∙[ψ*∇ψ - ψ∇ψ*] = -∇∙j.
  j = (ħ/(2im)[ψ*∇ψ - ψ∇ψ*] = (ħ/m) Im(ψ*∇ψ) = (1/m)Re(ψ*(ħ/i)∇ψ),
  We interpret ρ(r,t) as the probability density and j(r,t) as the probability current density.

Problem:

The classical Hamiltonian of a particle of charge q in the presence of a static magnetic field B is
H = (1/(2m)) (p - qA(r,t))².
The Schroedinger equation in quantum mechanics in coordinate representation is
Hψ(r,t) = (iħ∂/∂t)Ψ(r,t).
Find the differential operator for H in coordinate representation for a spin-less particle of charge q in the presence of a static magnetic field B.

Solution:

• Concepts:
  The gradient operator
• Reasoning:
  The gradient operator
  p = (ħ/i)∇ is a differential operator in coordinate space.
• Details of the calculation:
  H = (1/(2m)) (p - qA(r,t))².
  (p - qA)∙(p - qA)ψ = (ħ²∇²ψ + q²A²ψ - (ħq/i)∇∙A(r,t)ψ - (ħq/i)A∙∇ψ
  = -ħ²∇²ψ + q²A²ψ - (ħq/i)A∙∇ψ - (ħq/i)ψ∇∙A -( ħq/i)A∙∇ψ
  = -ħ²∇²ψ + q²A²ψ - (2ħq/i)A∙∇ψ, since ∇∙A = 0 in magnetostatics.
  H = (-ħ²/(2m))∇² + (q²/(2m))A²  + (iħq/m)A∙∇.