For states of definite energy
in quantum physics, the probability distribution is independent
of time: P(r) = |ψ(r,t)|2. Therefore we
should be able to describe a one-dimensional system with
definite energy entirely in terms of a state function ψ(x).
These are called “stationary states.”
The Schrödinger
equation for a particle of mass m in a potential V(x).
If we imagine a system with one degree of freedom, between
impenetrable walls at x = 0 and x = L, the
state functions must correspond to standing waves of
probability, with wavelength 2L, or L, or L/2,
etc. In other words, we set V = 0 between the walls,
so if we let ψ be proportional to sin(kx), and plug
into the equation, we require k2 = 2mE/ℏ2.
Now we impose the condition for standing waves! kn
= 2π/λn.
ψn(x)
= (2/L)1/2sin(nπx/L).
THE QUANTUM HARMONIC
OSCILLATOR
Solving the Schrödinger
equation gives energies En = ℏω[n + (1/2)], for U(x)
= (1/2)mω2x2. Here n is the
set of integers from 0 to infinity, and is called the
principal quantum number.
Quantum Tunneling! A
rough estimate of the probability that a particle of
energy E and mass m penetrates a rectangular
barrier with height Uo and width a
is given by P ≅ exp(-2α a) where α is the square
root of 2m(Uo - E)/ℏ2.
Note how the harmonic
oscillator state functions stick out of the potential.
They cannot tunnel out of it, but they can extend far out
into the classically forbidden region where the kinetic
energy becomes a negative number!
