5. The Harmonic Oscillator Flashcards
What is good about the Harmonic Oscillator and the TISE?
It is exactly solvable for potential
Describe u(x) and V(x) for the harmonic oscillator
u(x) - wave funciton
v(x) - 1/2 k’ x^2
Describe the difference between the classical potential and the harmonic oscillator (V(x))
- Classically, V(x) is the spring constant
- Now it is the curvature of the potential well
State the solutions for the wave functions for the harmonic well
u_n(x) = c_n sin( (n+1) pi*x/L)
What is the solution called when n = 0 for the harmonic well?
The ground state
State the ground state solution for the wave functions for the harmonic well
u_0(x) = c_0 sin(pi*x/L)
What should the ground state solution for the harmonic oscillator look like?
- Symmetrical
- Peak at 0
- Tend to 0 for x -> ± infinity
Page 3 on document
What form does the harmonic oscillator shape have for the ground state?
A Gaussian
State the Gaussian solution for the harmonic oscillator ground state
u_0 = c_0 exp(-α x^2)
Which SE do you use to solve the Gaussian shape for the ground state of the harmonic oscillator?
TISE and equate the squared and non squared terms
What solution is obtained for the ground state energy by solving the TISE with the Gaussian
E_0 = 1/2 h_bar w
How can you find c_0 from the Gaussian?
By integrating the modulus squared of u_0 over all space and setting it equal to 1
Describe u_0 and u_1 for the harmonic oscillator for 0 < x < L
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
State the trial solution for u_1(x) for the harmonic oscillator
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
What solution is obtained when solving for the n=1 state of the harmonic oscillator?
E_1 = 3/2 h_bar w
State the general energy solution for the harmonic oscillator
E_n = (n + 1/2) h_bar w
State the eigenvalue spectrum for the harmonic oscillator
H_hat u_n(x) = E_n u_n(x)
u_n(x) - eigen function
H_hat - Hamiltonian (energy operator)
E_n - eigenvalue
What is the classical assumption about the ground state?
That it has 0 KE and minimum (0) PE
What does the classical assumption about the ground state violate?
The H.U.P
Why is the ground state energy non-0?
Because of the “fuzziness” in x and p_x
What conditions must be met if we want to approximate any potential well?
- Must be sensible
- Use a parabolic potential provided we stay close to the minimum (small oscillation)
Describe the peak and trough of a potential graph
Peak - unstable equilibrium
Trough - stable equilibrium
What assumption can we make about diatomic molecules?
They are two masses connected by a spring
How do we model the separation of the two masses of a diatomic molecule?
x_0 - the “nuclear” separation
How are the longitudinal oscillations described for a diatomic molecule?
By the TISE solution - A “ladder” of quantum states and wave functions
Describe how the energy eigenvalue spectrum changes for the diatomic molecule
E_n = (n+1/2) h_bar w
but now w = sqrt(k’/μ)
here k’ - spring constant
μ - reduced mass = m1m2/m1+m2
What is μ for the diatomic molecule?
The reduced mass = m1m2/m1+m2
What is k’ for the diatomic molecule and what does it reflect?
The spring constant - reflects the strength of the chemical bond
Describe how we can “see” the molecular vibrations
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
Describe how isotopes would change the components of the diatomic molecule energy eigenvalue spectrum
One of the masses change, so the reduced mass μ changes. The chemical bonding is NOT changed, hence k’ is the same.
