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.
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 therefore will first complete Lab 3, Complex Numbers.
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) = ∑-∞+∞Cnexp(iknx),
Here kn = n2π/L, ∆k = kn+1 - kn = 2π/L. The coefficients Cm are given by
Cm = (1/L)∫0Lf(x)exp(-ikmx)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.