### 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
(-c2ħ22 + m2c4)ψ = -(ħ22/∂t2)ψ, or
(2 - m2c22)ψ = (1/c2)(∂2/∂t2)ψ.

#### Problem:

Consider the one-particle Schroedinger equation.
(a)  Write down an equation which states that probability is conserved.
(b)  By assuming that probability is locally conserved, derive an expression for the probability current density.

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 - ψψ*] = -∇∙j.
j
= (ħ/(2im)[ψ*ψ - ψψ*] = (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: