4. Wave Functions in 1D

Question 1

The TISE potential wells problem can be solved by using exp(ikx) solutions. Describe the system when E-V is >0 and <0

Answer 1

E-V > 0: exp(ikx) is oscillatory e.g sin(kx) etc

E-V < 0: exponential decay

Question 2

Describe the infinite potential well TISE curves for a particle in a box of length L

Answer 2

n = 1: half sine wave
n = 2: sine wave
n = 1: 3/2 sine wave
Picture 3 on document

Question 3

Describe the solution for the energies for the TISE for a particle in a box

Answer 3

E_n = (n^2) (pi)^2 (h bar)^2 / (2mL^2)
E_n - Energy eigenvalues (discrete energies)
n - quantum number - 1,2,3, etc

Question 4

Can the particle in a box have a solution when E = 0 and why?

Answer 4

No as it would have 0 momentum and KE. This would violate the H.U.P as Δx ~ L, Δp_x = 0

Question 5

Describe the finite potential well TISE curves for a particle in a box of length L

Answer 5

Same as infinite potential well, but the wavefunction does not equal 0 outside the boundary which is classically forbidden.

Question 6

How can we calculate the probability density of the wave function?

Answer 6

By taking the squared modulus of the wave function

Question 7

What type of wave is the wave function?

Answer 7

Continuous

Question 8

For a finite step in potential, what can be said about the gradient of the wave function?

Answer 8

It is also continuous

Question 9

Compare the assumptions for the wave function when it is spread over a large, and short distance

Answer 9

Large distance - well defined momentum
Short distance (squashed) - not a well defined momentum

Question 10

State the assumptions that are a consequence of the H.U.P

Answer 10

Assumptions for the wave function spread over distances:
Large distance - well defined momentum
Short distance (squashed) - not a well defined momentum

Question 11

State the H.U.P

Answer 11

ΔxΔp >= h_bar / 2

Question 12

Describe the H.U.P graphs for position and momentum over different wave function distances

Answer 12

• Delta position -> continuous momentum over large distance - not well defined
  Delta momentum, continuous position over large distance - well defined
  Picture 4 on document

Question 13

What can be said when Δk gets smaller

Answer 13

The wave is closer to a sine or cos wave - the wave packet is less localised

Question 14

How can Gaussian wave packets be built?

Answer 14

By taking the Gaussian distribution of wave vectors