5. The Harmonic Oscillator

Question 1

What is good about the Harmonic Oscillator and the TISE?

Answer 1

It is exactly solvable for potential

Question 2

Describe u(x) and V(x) for the harmonic oscillator

Answer 2

u(x) - wave funciton

v(x) - 1/2 k’ x^2

Question 3

Describe the difference between the classical potential and the harmonic oscillator (V(x))

Answer 3

• Classically, V(x) is the spring constant

- Now it is the curvature of the potential well

Question 4

State the solutions for the wave functions for the harmonic well

Answer 4

u_n(x) = c_n sin( (n+1) pi*x/L)

Question 5

What is the solution called when n = 0 for the harmonic well?

Answer 5

The ground state

Question 6

State the ground state solution for the wave functions for the harmonic well

Answer 6

u_0(x) = c_0 sin(pi*x/L)

Question 7

What should the ground state solution for the harmonic oscillator look like?

Answer 7

• Symmetrical
• Peak at 0
• Tend to 0 for x -> ± infinity
  Page 3 on document

Question 8

What form does the harmonic oscillator shape have for the ground state?

Answer 8

A Gaussian

Question 9

State the Gaussian solution for the harmonic oscillator ground state

Answer 9

u_0 = c_0 exp(-α x^2)

Question 10

Which SE do you use to solve the Gaussian shape for the ground state of the harmonic oscillator?

Answer 10

TISE and equate the squared and non squared terms

Question 11

What solution is obtained for the ground state energy by solving the TISE with the Gaussian

Answer 11

E_0 = 1/2 h_bar w

Question 12

How can you find c_0 from the Gaussian?

Answer 12

By integrating the modulus squared of u_0 over all space and setting it equal to 1

Question 13

Describe u_0 and u_1 for the harmonic oscillator for 0 < x < L

Answer 13

u_0 - even function about L/2. 1 Maxmimum
u_1 - odd function about L/2. 2 Maxima
Page 3 on document, but shift the curve from -L/2 < x < L/2 to 0 < x < L

Question 14

State the trial solution for u_1(x) for the harmonic oscillator

Answer 14

u_1(x) = c_1 (x)xexp(-α x^2)

Multiply by a pre factor of x (odd) to generate the shape on page 3 of document

Question 15

What solution is obtained when solving for the n=1 state of the harmonic oscillator?

Answer 15

E_1 = 3/2 h_bar w

Question 16

State the general energy solution for the harmonic oscillator

Answer 16

E_n = (n + 1/2) h_bar w

Question 17

State the eigenvalue spectrum for the harmonic oscillator

Answer 17

H_hat u_n(x) = E_n u_n(x)
u_n(x) - eigen function
H_hat - Hamiltonian (energy operator)
E_n - eigenvalue

Question 18

What is the classical assumption about the ground state?

Answer 18

That it has 0 KE and minimum (0) PE

Question 19

What does the classical assumption about the ground state violate?

Answer 19

The H.U.P

Question 20

Why is the ground state energy non-0?

Answer 20

Because of the “fuzziness” in x and p_x

Question 21

What conditions must be met if we want to approximate any potential well?

Answer 21

• Must be sensible

- Use a parabolic potential provided we stay close to the minimum (small oscillation)

Question 22

Describe the peak and trough of a potential graph

Answer 22

Peak - unstable equilibrium

Trough - stable equilibrium

Question 23

What assumption can we make about diatomic molecules?

Answer 23

They are two masses connected by a spring

Question 24

How do we model the separation of the two masses of a diatomic molecule?

Answer 24

x_0 - the “nuclear” separation

Question 25

How are the longitudinal oscillations described for a diatomic molecule?

Answer 25

By the TISE solution - A “ladder” of quantum states and wave functions

Question 26

Describe how the energy eigenvalue spectrum changes for the diatomic molecule

Answer 26

E_n = (n+1/2) h_bar w
but now w = sqrt(k’/μ)
here k’ - spring constant
μ - reduced mass = m1m2/m1+m2

Question 27

What is μ for the diatomic molecule?

Answer 27

The reduced mass = m1m2/m1+m2

Question 28

What is k’ for the diatomic molecule and what does it reflect?

Answer 28

The spring constant - reflects the strength of the chemical bond

Question 29

Describe how we can “see” the molecular vibrations

Answer 29

The gap between states E_n and E_n+1 is the vibrational energy level.
A photon of energy h_bar w is absorbed and the molecule “jumps” to the next energy level

Question 30

Describe how isotopes would change the components of the diatomic molecule energy eigenvalue spectrum

Answer 30

One of the masses change, so the reduced mass μ changes. The chemical bonding is NOT changed, hence k’ is the same.