(a) Define what is meant by
the term “stationary state” in quantum mechanics, and explain the distinction
between the time-dependent and time-independent Schrödinger equation.

(b) At time t = 0, the wave
function of a particle in one dimension is ψ(x) = (u_{1}(x)
+ u_{2}(x))/√2, where u_{1}(x) and u_{2}(x) are
two solutions of the time-independent Schrödinger equation.

For this particle,
how does the probability density change with time?

In relativistic mechanics, energy and momentum of a free
particle are related by the expression E^{2} = p^{2}c^{2}
+ m^{2}c^{4}. 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:

(p^{2}c^{2}+ m^{2}c^{4})ψ = E^{2}ψ

The momentum operator is (ħ/i)**∇**. The energy operator is iħ∂/∂t.

We therefore have

(-c^{2}ħ^{2}**∇**^{2}+ m^{2}c^{4})ψ = -(ħ^{2}∂^{2}/∂t^{2})ψ, or

(**∇**^{2}- m^{2}c^{2}/ħ^{2})ψ = (1/c^{2})(∂^{2}/∂t^{2})ψ.

(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 d^{3}r about**r**is given by dP(**r**,t) = |ψ(**r**,t)|^{2}d^{3}r. The total probability of finding the particle at time t anywhere in space is ∫_{all space}|ψ(**r**,t)|^{2}d^{3}r = <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*((-ħ^{2}/(2m))∇^{2}ψ + Vψ) - ((-ħ^{2}/(2m))∇^{2}ψ + Vψ)*ψ]

= (-ħ/(2im)[ψ*∇^{2}ψ -(∇^{2}ψ)*ψ], since V is real.

∂ρ/∂t = (-ħ/(2im)**∇∙**[ψ***∇**ψ - ψ**∇**ψ*] = -**∇∙j.**= (-ħ/(2im)[ψ*

j**∇**ψ - ψ**∇**ψ*] = (-ħ/(m) Im(ψ***∇**ψ).

We interpret ρ(**r**,t) as the probability density and**j**(**r**,t) as the probability current density.