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 Schrödinger 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 Schrödinger 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:

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: