PDF with typeset equations

Quantum mechanics lets us predicts the behavior of all particles using a wave equation.  It lets us calculate a wave function for each particle.  This wave function, at a particular time, contains all the information that anybody at that time can have about the particle, such as information about its location, its momentum, its energy, its angular momentum, etc.  Quantum mechanics deals with two problems.

• How do we determine the wave function ψ(x,t) of a particle?
• How do we extract information about the particle from the wave function?

We will first look at the problem of finding the wave function.  The wave functions of particles are all solutions of a wave equation.  The wave equation that tells us how matter waves evolve is the Schroedinger equation.
The Schroedinger equation for a particle with mass m moving in one dimension has the form

 (-ħ²/(2m))∂²ψ(x,t)/∂x² + U(x)ψ(x,t)  = iħ∂ψ(x,t)/∂t.

Here U(x) is the potential energy of the particle when it is at position x.  The function U(x) contains the information about the particle's environment.  For example, if the particle is attached to a spring and its equilibrium position is x = 0, then its potential energy at position x is U(x) = (1/2)kx².

The wave function ψ(x,t) of each particle is a solution of the Schroedinger equation.  ψ(x,t) contains all the information an observer can possibly obtain about the particle.

For a free particle with mass m, such as an electron moving freely through space, U(x) = 0, and the Schroedinger equation is of the form

(-ħ²/(2m))∂²ψ(x,t)/∂x² = iħ∂ψ(x,t)/∂t.

The Schroedinger equation is an equation involving complex numbers.  To be able to investigate and understand its solutions, we need an introduction to complex numbers.  We will cover this in greater detail in the next class period.

Link: A complex Numbers Tutorial

Aside:  Solutions to the wave equation for waves on a string

For the wave equation for a wave on a string,

∂²y/∂x² - (1/v²)∂²y/∂t² = 0,

we have found two linearly independent solution, y(x,t) = Acos(kx - ωt) and y(x,t) = Bsin(kx - ωt) with |v| = ω/k.  Any other solution is a linear combination of those solutions.  The functions  y(x,t) = exp(i(kx - ωt)) and y(x,t) =  exp(-i(kx - ωt))  are linear combinations  of cos(kx - ωt) and  sin(kx - ωt).

exp(i(kx - ωt))  = cos(kx - ωt) + i sin(kx - ωt).
exp(-i(kx - ωt))  = cos(kx - ωt) - i sin(kx - ωt).

They are therefore also general solutions of this wave equation and can be used to expand any specific solution.  They can be used to build wave packets.   

This leads to a different way of expressing the Fourier Series and the Fourier transform.  Any periodic (and reasonably continuous) function with spatial period (wavelength) L can be synthesized by a sum of imaginary exponentials whose spatial periods (wavelengths) are integral submultiples of L, (such as L/2, L/3, ...). 
Let f(x) be such a periodic function.  Then we may write

f(x) = ∑_-∞^+∞C_nexp(ik_nx),

Here k_n = n2π/L, ∆k = k_n+1 - k_n = 2π/L.  The coefficients C_m are given by

C_m = (1/L)∫₀^Lf(x)exp(-ik_mx)dx.

When L becomes very large, i.e. L --> ∞, the Fourier series generalizes to the Fourier transform.  

Solutions to the Schroedinger equation for a free particle

The functions ψ(x,t) = exp(i(kx - ωt)) are general solutions to the Schroedinger equation for a free particle,

(-ħ²/(2m))∂²ψ(x,t)/∂x² = iħ∂ψ(x,t)/∂t.

We call these solutions plane wave solutions.  If ψ(x,t) = exp(i(kx - ωt)), then

∂²ψ(x,t)/∂x² = -k²exp(i(kx - ωt)) = -k²ψ(x,t),     ∂ψ(x,t)/∂t = -iω exp(i(kx - ωt))) = -iωψ(x,t).

Therefore

(-ħ²/(2m))∂²ψ(x,t)/∂x² = (ħ²k²/(2m))ψ(x,t)
iħ∂ψ(x,t)/∂t = ħωψ(x,t).

We need ħ²k²/(2m) = ħω for ψ(x,t) = exp(i(kx - ωt)) to be a solution of the Schroedinger equation for a free particle.

The deBroglie relations for a free particle are λ = h/p and f = E/h or p = ħk and E = ħω.
Therefore the expression ħ²k²/(2m) = ħω is equivalent to the expression p²/(2m) = E. 
Using the deBroglie relations we can write the wave function ψ(x,t) = exp(i(kx - ωt)) as ψ(x,t) = exp(i(px - Et)/ħ).
This wave function ψ(x,t) = exp(i(px - Et)/ħ) represents a particle with a definite momentum p and a definite energy E.

The wave function ψ(x,t) = exp(i(kx - ωt)) is a solution of the Schroedinger equation for a free particle if ħ²k²/(2m) = ħω.  Any constant A will work, and any wave number k will work, but if we pick k, then ω is fixed.  A positive k represents a wave traveling into the positive x-direction and a negative k represents a wave traveling into the negative x-direction.
Any linear superposition of solutions of these two types is also a solution to the Schroedinger equation for a free particle.

[Note: You can verify by direct substitution that ψ(x,t) = Acos(kx - ωt) and ψ(x,t) = Bsin(kx - ωt) are not solutions of the Schroedinger equation.  The Schroedinger equation does not have real, but complex solutions.]

So which solution is the right solution for a particular situation?  We can build traveling wave packets by superimposing plane waves with many slightly different wave numbers k, or construct standing waves by superimposing waves traveling into opposite directions.  What do we need to do?  We are guided by the way we interpret the wave function.