The operator representing the energy of a system is H. The
eigenvalues of H are
E. If the potential
energy function U(x) is
independent of time, then separation of variables is possible, and we can write
ψ(x,t) = Φ(x)Χ(t).
If the wave function is of this form, then Φ(x) =
ΦE(x)
is an eigenfunction of the operator H, and the energy of the system is
certain. We find the eigenfunction of H by solving
HΦE(x) = EΦE(x).
Regions that do not contain a well
There exists an eigenfunction for every E > Umin. These
eigenfunctions, however, are plane waves and are not square integrable.
They cannot represent a single particle, but can represent a constant flux
of particles. We calculate transmission and reflection coefficients by
comparing fluxes. (Flux =
k|Φ(x)|2.)
Regions that do contain a well
For E < Erim eigenfunctions exist only for selected
eigenvalues. These eigenfunctions are square integrable.
Confinement leads to energy quantization.
We can solve HΦE(x) = EΦE(x) in regions of
piecewise constant potentials.
Assume U(x) = U = constant in certain regions of space. In such a region the Schroedinger equation yields
(∂2/∂x2)Φ(x)
+ (2m/ħ2)(E
- U)Φ(x)
= 0.
i. Let E > U: (∂2/∂x2)Φ(x)
+ k2Φ(x)
= 0. E - U = ħ2k2/(2m).
The most general solution is
Φ(x) = Aexp(ikx) + A'exp(-ikx), with A and A'
complex constants.
ii. Let E < U: (∂2/∂x2)Φ(x)
- ρ2Φ(x) = 0.
U - E = ħ2ρ2/(2m).
The most general solution is
Φ(x) = Bexp(ρx)
+ B'exp(-ρx),
with B and B' complex constants.
iii. Let E = U: (∂2/∂x2)Φ(x)
= 0. Φ(x)
= Cx + C', with C and C' complex constants.
(Note: A solution exists in the classically forbidden regions.)
How does the wave function behave at a point where U is discontinuous, i.e.
at a step?
(a) At a finite step the boundary conditions are that
Φ(x) and (∂/∂x)Φ(x)
are continuous.
(b) At an infinite step (∂/∂x)Φε(x)|ε-->0
is discontinuous, but
it has a finite discontinuity. Therefore
Φε(x) remains continuous as
ε -->
0.
We can solve for the bound states in a square-well potential using a graphical solution.
H = -(ħ2/(2m))(∂2/∂x2)
+ ½mω2x2
is the harmonic oscillator Hamiltonian.
(∂2/∂x2)Φ(x)
+ (2m/ħ2)(E
- ½mω2x2)Φ(x)
= 0 is the eigenvalue equation.
The eigenvalues are En
= (n + ½)ħω,
n = 0, 1, 2, ... .
The normalized ground state wave
function of the 1-dimensional harmonic oscillator is
Φ0(x)
= (mω/(πħ))¼exp(-½mωx2/ħ).
The wave functions of the excited states are
Φn(x) = (n! 2n)-½(β/√π)½Hn(η)
exp(-½η2),
where η = (mω/ħ)½ x = βx.
Specifically,
Φ1(x) = ((4/π)(mω/ħ)3)¼ x exp(-½mωx2/ħ),
Φ2(x) = (mω/(4πħ))¼[2mωx2/ħ
- 1] exp(-½mωx2/ħ).
Φn(x) is the product of exp(-½mωx2/ħ) and a polynomial of degree n
and parity (-1)n called a
Hermite
polynomial.
Defining scaled operators
Xs = (mω/ħ)½X, Ps = (mωħ)-½P,
and new operators
a = (2)-½(Xs + iPs), and its adjoint, a†
= (2)-½(Xs - iPs), we can write
H = ħω(a†a + ½) = ħω(aa† - ½) =
½ħω(aa† + a†a).
The operators a and a† do not commute.
[a,a†] = 1.
H and a†a have the same eigenstates {|n>}.
a†a|n> = n|n>, H|n> = (n + ½)ħω|n>.
a|n> = √(n) |n-1>, a†|n> = √(n+1) |n+1>.