Hamonic Oscillator-Classical Case in 1D Flashcards
What is the equation for the conservative force in 1D for a harmonic oscillator?
F(x) = -k’x, where k’ is the spring constant and k is the wavenumber
What is the equation for the hamonic potential in 1D for a harmonic oscillator?
V(x) = 1/2 * k’ *x^2
On a graph of V against x, what are the turning points?
Where F = -dV/dx = 0
On a graph of V against x, when are the turning points stable?
if d^2V/dx^2 > 0, is stable, unstable otherwise
For a stable equilibrium, how can we approximate the bottom of the potential well?
With a harmonic oscillator: V(x) ~ V(a) + 1/2 * k(a)’ *(x-x(a))^2
How can we approximate this equation for the harmonic oscillator?
V(x(a)) -> 0, and x-x(a) -> 0, so we end up with the normal equation for the harmonic potential.
On a V-x graph for an O2 molecule, what can we approximate the turning point region to?
Approximate to a parabola, so diatomic molecule will oscillate between some limits x= +/- a
What is the time independent schrodinger equation?
-ћ/2m * d^2u/dx^2 + Vu = Eu, where u = u(x) = wavefunction
What do we do to the TISE to get an equation for d^2u/dx^2?
Substitute in the equation for V(x) and rearrange.
What is the equation for the solutions for the flat well? (u(x))
u(x) = C*sin(nπx/L)
How can we solve the equation for d^2u/dx^2?
Try a gaussian solution: u(x) = c*exp(-αx^2). Sub this in.
What do we do once we have solved d^2u/dx^2?
Equate terms in x^2u and constant * u, to find equations
What do we find for the equation of E after equating? What is the equation for u0(x)?
E = E0 = 1/2 * ћω, u0(x) = C0exp(-mωx^2/2ћ)
What is the trial solution for the first excited state u1?
Odd function: u1(x) = C1xexp(-αx^2)
What do we do with the trial solution for u1 and what do we get for E1?
Sub into the TISE, and we get E1 = 3/2 ћω
What is the general trial solution for u(n)?
CnHn(x)exp(-αx^2)
When does the trial solution for u(n) work?
When Hn(x) are Hermite polynomials: n=0, Hn(x) = 1, n=1, Hn(x) = 2x, n=2, Hn(x) = 4x^2 -2, n=3, Hn(x) = 8x^3 - 12x
How does En increase with n?
n=0, En = 1/2 ћω, n=1, En = 3/2 ћω, etc
What do the En values form?
The eigenvalue spectrum.
What is ΔE equal to?
ΔE = En+1 - En = ћω = ћsqrt(k’/m)
What do we do to try and understand the zero-point energy E0 = 1/2 ћω?
Find E0 classically: E0 = ρ^2/2m + V(x), V(x) = mω^2 * x^2/2
What do we do with the classical equation for E0?
Slow everything down: ρ -> 0, x -> 0, V(x) -> 0, E0 -> 0
Why is the QM case for E0 different to the classical case?
Use HUP and sub in the rearranged equation for ρ.
What is Heisenbergs uncertainty principle?
ΔxΔρ >= ћ/2
How can we minimise E0 after subbing in HUP?
via dE0/dΔx = 0, so compute this differential and set equal to zero
What equation do we get after computing the differential and rearranging? (for Δx)
Δx^2 = ћ/2mω
What do we do with this equation for Δx?
Plug back into the equation for E0 with HUP subbed in -> find that this is equal to 1/2 ћ*ω, the zero-point energy
