Hamonic Oscillator-Classical Case in 1D

Question 1

What is the equation for the conservative force in 1D for a harmonic oscillator?

Answer 1

F(x) = -k’x, where k’ is the spring constant and k is the wavenumber

Question 2

What is the equation for the hamonic potential in 1D for a harmonic oscillator?

Answer 2

V(x) = 1/2 * k’ *x^2

Question 3

On a graph of V against x, what are the turning points?

Answer 3

Where F = -dV/dx = 0

Question 4

On a graph of V against x, when are the turning points stable?

Answer 4

if d^2V/dx^2 > 0, is stable, unstable otherwise

Question 5

For a stable equilibrium, how can we approximate the bottom of the potential well?

Answer 5

With a harmonic oscillator: V(x) ~ V(a) + 1/2 * k(a)’ *(x-x(a))^2

Question 6

How can we approximate this equation for the harmonic oscillator?

Answer 6

V(x(a)) -> 0, and x-x(a) -> 0, so we end up with the normal equation for the harmonic potential.

Question 7

On a V-x graph for an O2 molecule, what can we approximate the turning point region to?

Answer 7

Approximate to a parabola, so diatomic molecule will oscillate between some limits x= +/- a

Question 8

What is the time independent schrodinger equation?

Answer 8

-ћ/2m * d^2u/dx^2 + Vu = Eu, where u = u(x) = wavefunction

Question 9

What do we do to the TISE to get an equation for d^2u/dx^2?

Answer 9

Substitute in the equation for V(x) and rearrange.

Question 10

What is the equation for the solutions for the flat well? (u(x))

Answer 10

u(x) = C*sin(nπx/L)

Question 11

How can we solve the equation for d^2u/dx^2?

Answer 11

Try a gaussian solution: u(x) = c*exp(-αx^2). Sub this in.

Question 12

What do we do once we have solved d^2u/dx^2?

Answer 12

Equate terms in x^2u and constant * u, to find equations

Question 13

What do we find for the equation of E after equating? What is the equation for u0(x)?

Answer 13

E = E0 = 1/2 * ћω, u0(x) = C0exp(-mωx^2/2ћ)

Question 14

What is the trial solution for the first excited state u1?

Answer 14

Odd function: u1(x) = C1xexp(-αx^2)

Question 15

What do we do with the trial solution for u1 and what do we get for E1?

Answer 15

Sub into the TISE, and we get E1 = 3/2 ћω

Question 16

What is the general trial solution for u(n)?

Answer 16

CnHn(x)exp(-αx^2)

Question 17

When does the trial solution for u(n) work?

Answer 17

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

Question 18

How does En increase with n?

Answer 18

n=0, En = 1/2 ћω, n=1, En = 3/2 ћω, etc

Question 19

What do the En values form?

Answer 19

The eigenvalue spectrum.

Question 20

What is ΔE equal to?

Answer 20

ΔE = En+1 - En = ћω = ћsqrt(k’/m)

Question 21

What do we do to try and understand the zero-point energy E0 = 1/2 ћω?

Answer 21

Find E0 classically: E0 = ρ^2/2m + V(x), V(x) = mω^2 * x^2/2

Question 22

What do we do with the classical equation for E0?

Answer 22

Slow everything down: ρ -> 0, x -> 0, V(x) -> 0, E0 -> 0

Question 23

Why is the QM case for E0 different to the classical case?

Answer 23

Use HUP and sub in the rearranged equation for ρ.

Question 24

What is Heisenbergs uncertainty principle?

Answer 24

ΔxΔρ >= ћ/2

Question 25

How can we minimise E0 after subbing in HUP?

Answer 25

via dE0/dΔx = 0, so compute this differential and set equal to zero

Question 26

What equation do we get after computing the differential and rearranging? (for Δx)

Answer 26

Δx^2 = ћ/2mω

Question 27

What do we do with this equation for Δx?

Answer 27

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