Harmonic oscillator

Question 1

Classical Hooke’s law

Answer 1

F = -kx, where;
F is force, Newtons
x is distance, m
k is the spring constant#
- value as it is a restoring force

Question 2

Newtons 2nd law and Hooke’s law:

Answer 2

F=ma=m(d^2x/dt^2)=-kx

Question 3

Energy stored in vibration

Answer 3

V = ∫-Fdx=∫kxdx=1/2kx^2

Question 4

Harmonic spring general equation

Answer 4

x = Asin(wt+phi)
Where A is a constant, w is the angular velocity (rads^-1) and phi is phase

Question 5

Force with harmonic equation

Answer 5

F=ma=m(d^2x/dt^2)=-mw^2Asin(wt+phi)=-mw^2x=-kx

Question 6

Angular velocity as a function of spring constant and mass

Answer 6

w = (k/m)^1/2, found from force with harmonic

Question 7

Frequency of vibration

Answer 7

v^ = w/2pi = 1/2pi*(k/m)^1/2

Question 8

Boundary condition

Answer 8

no infinitely large compressions or extensions

Question 9

Consequence of boundary

Answer 9

E = (v+1/2)hv^=(v+1/2)ℏw where v is a quantum number v = 0,1,2,3…
v^ is the vibrational frequency and w is angular velocity

Question 10

Is vibrational energy quantised

Answer 10

Yes as v is a quantum number v=0,1,2,3… and is a multiple of E

Question 11

Zero point for vibration:

Answer 11

Occurs at v = 0, E = 1/2ℏw,
not 0 as particle is confined

Question 12

Hv for v=0

Answer 12

1

Question 13

Hv for v=1

Answer 13

ζ

Question 14

Hv for v=2

Answer 14

4ζ^2 -2

Question 15

ζ =

Answer 15

x/a

Question 16

a^4 =

Answer 16

ℏ^2/km

Question 17

Ψ for harmonic oscillator

Answer 17

Ψ =Hv*e^(ζ^2/2)

Question 18

How to help determine wavefunction

Answer 18

Rearrange Schrodinger’s for d^2Ψ/dx^2 + rest = 0

Question 19

What happens for larger masses and stiffer strings?

Answer 19

Wavefunction will decay more rapidly due to ζ dependence on k and m

Question 20

What happens to the behaviour of the probability distribution as v (Qnum) increases

Answer 20

Position of high probability will migrate towards turning points, more like classical mechanics