Three dimensions:
Three-dimensional infinite square well:
E = Ex + Ey + Ez. E = nx2π2ħ2/(2mLx2)
+ ny2π2ħ2/(2mLy2) +
nz2π2ħ2/(2mLz2).
Orbital angular momentum:
In three dimensions, for a potential energy function U(x,y,z) = U(r), the
solutions of the time-independent Schroedinger equation are
ψnlm(r,θ,ϕ) = Rnl(r)Ylm(θ,ϕ).
The spherical harmonics Ylm(θ,ϕ)
are eigenfunctions of the operator L2, which is the operator for the
square of the magnitude of the orbital angular momentum.
L2Ylm(θ,ϕ)
= l(l+1)ħ2Ylm(θ,ϕ), l = 0, 1, 2, ... .
The magnitude of the angular momentum is quantized. The only possible outcomes
of a measurement are (l(l +
1)ħ2.
The Ylm(θ,ϕ)
are also eigenfunctions of the operator Lz, which is the operator for
the z-component of the orbital angular momentum.
LzYlm(θ,ϕ) =
mħYlm(θ,ϕ), m = 0, ±1, ±2,
... ±l.
The z-component of the angular momentum is quantized. The only possible
outcomes of a measurement are
mħ.
Tunneling:
The transmittance T is the probability that a particle will tunnel through a
barrier.
T = exp(-2bL) with b = (2m(U0-E)/ħ2)1/2.
T decreases as the difference between the height of the barrier U0
and the particle energy increases, and it is smaller for a more massive particle
and for a wider barrier.