Quantum-mechanical Superposition Exercises

 

Time-dependent quantum-mechanical wave functions are inherently complex (having real and imaginary components) due to the time evolution governed by the Schrödinger equation, which in one-dimensional position space is

[−(ħ2/2m)∂2/∂x2 + V(x)] ψ(x,t) = (∂/∂t) ψ(x,t) .      


The standard way to visualize the wave functions that solve this equation is to either consider just the real part or consider the probability density, approaches that discard all phase information.  We use color-as-phase representation of the wave function to display this information in a meaningful way.

States with non-trivial time evolution have non-trivial time evolution.  The equal-mix two-state superposition is one of the simplest examples of non-trivial time-dependent states.  These position- and momentum-space wave functions are just

Ψn1n2(x,t) = 2−1/2n1(x,t) + ψn2(x,t)] ,

and

Φn1n2(p,t) = 2−1/2n1(p,t) + φn2(p,t)] .

We can examine the time dependence of more general states by adopting a general description for a time-dependent superposition:

Ψ(x,t) = Σ cn exp[−iEnt/ħ] ψn(x)        [sum from n = 1 to n = ∞] .  

where the expansion coefficients satisfy Σ  |cn|2 = 1.  Such a time-dependent state in the infinite square well experiences so-called quantum-mechanical revivals, in which an initially localized wave packet reforms a definite time later.

© Mario Belloni and Wolfgang Christian (2006).