Section 5 Flashcards
If the energy of the state is lower than the bound energy then
it will be bound
Specifically, tunnelling of wave-functions through finite-width barrier means we define a state as bound if
E < V(±∞)
In general a potential can have both
bound and scattering states
Particle inside a box: infinite square-well
inside the well, behaves like a free particle
-ℏ/2m d^2Ψ(x)/dx^2 = EΨ(x)
with k = √(2mE) / ℏ
infinite square-wells have the general solution most conveniently express as
Ψ(x) = Acoskx + Bsinkx
infinite square-wells have boundary conditions at
|x| = a => Ψ(±a) = 0
Boundary conditions quantize
eigenstates in even/odd pairs Ψ(x)
u(x)n+ =
1/√a cos nπx/2a
with n = 1,3, … ∞
u(x)n- =
1/√a sin nπx/2a
with n = 2,4, … ∞
Energies formula
En+ = n^2 π^2ℏ^2/8ma^2
u(x)1+ is the
ground state
Normalisation <un±|un±>
= 1
Orthogonality <un±|um∓>
= 0
<un±|um±> = δnm
Expectation values
<x> = 0
<p> = 0
<p^2> =2mEn±
</p></x>
The spatial eigenstates have definite
parity, i.e. reflection symmetry x -> -x
An asymmetric potential, V(x) ≠ V(-x), will not
have eigenstates of definite parity
The delta function well takes the form
d^2Ψ/dx^2 = -2mE/ℏ^2 Ψ = κ^2Ψ
for κ = √(-2mE)/ℏ
For the delta function well the general solution for real κ
Ψ(x) = Ae^(-κx) + Be^(κx)
Finite square well: eigenfunctions in the well are with
λ = √(2mE)/ℏ
Finite square well: in the walls exponential decay with
κ = √(-2m(E-V))/ℏ
solving the finite well:
Even => tan(λa) = κ/λ
Odd => -cot(λa) = κ/λ
Schrodinger Equation as a Simple Harmonic Oscillator
Quadratic well
-ℏ^2/2m d^2Ψ/dx^2 + 1/2mω^2x^2Ψ = EΨ
The Simple Harmonic Oscillator Hamiltonian is
H(hat) = [p(hat)^2 + (mωx(hat)^2]/2m
a(hat)± are the
simple harmonic oscillator ladder operators
