Quantum Mechanics

Question 1

What is the wave equation?

Answer 1

d^2 y/dx^2 = 1/v^2 * d^2 y/dt^2

Question 2

What can a general solution to the wave equation be written as?

Answer 2

y = Aexp(ikx-iωt)

Question 3

What is the time-dependent Schrodinger equation?

Answer 3

-ħ/2m * d^2 Ψ/dx^2 = iħ * dΨ/dt, where Ψ is a wave-function

Question 4

What would you get if you substitute the general solution into the TDSE?

Answer 4

Just differentiate it:

-ħ^2/2m(ik)^2exp(ikx-iωt) = iħ(-iω)exp(ikx-iωt)

Question 5

How do you find the solution to the TDSE?

Answer 5

Take the real parts and set equal to zero. Is a solution if certain conditions are met.

Question 6

What is the equation for energy when you combine Planck’s equation and de Broglies equation?

Answer 6

E = (ħ^2*k^2)/(2m) = p^2/2m = KE

Question 7

What is the meaning of Ψ?

Answer 7

Ψ is complex. Bohr suggested that |Ψ(x,t)|^2 dx = Ψ(x,t)Ψ(x,t) is the probability of finding a particle at time t along line element between x and x+dx -> |Ψ(x,t)|^2 is the probability density.

Question 8

What is the normalisation condition for Ψ?

Answer 8

integral over all relevant space of |Ψ(x,t)|^2 dx = 1

Question 9

What do you get if you use the normalisation condition for a plane wave? What does this mean?

Answer 9

|Ψ|^2 = ΨΨ* = A^2 * exp(ikx-iωt)*exp(-i(kx-ωt)) = A^2. This is not a good solution as it means the particle is equally likely to be anywhere.

Question 10

How can you find a good solution to the wavefunction?

Answer 10

Need to form wavepackets to represent particle - several examples of how the wave function can be combined.

Question 11

What is step one when considering wave packets?

Answer 11

Consider Ψ1 and Ψ2, where the TDSE applies to each separately. Then Ψ = αΨ1 + βΨ2

Question 12

What is the second step when considering wave packets?

Answer 12

• Consider 2 plane waves travelling in opposite directions : Ψ1 = Aexp(ikx-iω(k)t), Ψ2 = Aexp(-ikx-iω(-k)t)
• Add these together and find that it is a standing wave which oscillates with angular frequency Aexp(ikx-ω(k) and amplitude 2A cos(kx)

Question 13

Where is the particle when considering 2 waves in opposite directions?

Answer 13

|Ψ|^2 = 4A^2cos^2(kx) = 2A^2(1+cos(2kx)) - particle more likely to be found in some places than others - draw graph of amplitude against cos(theta) and sub in max and min values for cos

Question 14

What happens when you consider 2 plane waves travelling in the same direction but with slightly different properties?

Answer 14

Ψ1 = Aexp(ik1x-iω(k1)t), Ψ2 = Aexp(ik21x-iω(k2)t)

• Choose k1 = k0 + dk and k2 = k0 - dk
• Expand ω1 and ω2 to become ω0 +/- Δk*ω0’
• Put these into the equation for Ψ = Ψ1 + Ψ2
• Rearrange

Question 15

What does the probability density look like for two waves travelling in the same direction?

Answer 15

Similar to two waves travelling in opposite direction, but is a travelling wave moving at speed vg = ω0’

Question 16

What is the equation for the group velocity?

Answer 16

dω/dk = ω’ = group velocity

Question 17

How can we find the momentum using the group velocity?

Answer 17

Use ħω(k) = (ħ^2k^2)/(2m), so ω’ = d/dk(ħk^2)/(2m)) = (ħk)/(m), so ω0’ = (ħk0)/(m) = p0/m, where p0 is the average momentum of a particle

Question 18

What is the normalisation problem?

Answer 18

Plane waves, standing waves and travelling waves considered before cannot be normalised: integral between -infinity and infinity of |Ψ|^2 dx is infinity =/ 1 unless A = 0

Question 19

How can you solve the normalisation problem?

Answer 19

Integrate between -L and L instead of infinities. Integral between -L and L of A^2 dx = 2LA^2 = 1 -> A = 1/sqrt(2L) = normalisation condition.

Question 20

What are wave packets? What is the equation for a wave packet Ψ?

Answer 20

Superposition of many plane waves with wave numbers around an average value, k0. Assume continuous distribution of waves with amplitudes a(k). Equation is the integral of dka(k)exp(ikx-iω(k)t)

Question 21

What characterises the distribution of waves in wave packets?

Answer 21

The width of the curve when you plot a(the amplitude) against k.

Question 22

What is the equation for the width of the curve a-k?

Answer 22

w^2 = (integral of dk |a(k)|^2 * (k-k0))/(integral of dk (a(k))^2)

Question 23

What an you do if the width w is narrow (w &laquo_space;k0)? What does this give?

Answer 23

ω(k) = ω0 + ω0’ Δk. Substitute this into the equation for Ψ. This gives a solution for 2 waves combined.

Question 24

What is the probability density of Ψ (wave packet)?

Answer 24

Ψ depends of f(s) (function of x-ω0’t = s), so probability density is |Ψ|^2 = |f(s)|^2

Question 25

What happens if a(k) is a Gaussian of width w? What then happens if w is large?

Answer 25

The corresponding probability density is also a Gaussian of width 1/w which travels at speed vg = ω0’. If w is large, 1/w is small so P(x) is narrow and position of particle is well defined.

Question 26

How would you go about measuring the quantum uncertainty in a ball for example?

Answer 26

Use ΔpΔx >= ħ/2. Ignore the 1/2 and rearrange to find momentum uncertainty, then use mass to calculate the velocity uncertainty.

Question 27

What is the top hat distribution?

Answer 27

a(k) = A, k0 - w/2 < k < k0 + w/2. Then integrate between -w/2 and w/2 dk’ exp(ik’s). Gives a “top hat” distribution with width 2pi/w.

Question 28

What is the equation for the uncertainty principle?

Answer 28

ΔpΔx >= ħ/2

Question 29

Consider Bohr’s Hydrogen atom. What is the equation for Δp?

Answer 29

Δp >= ħ/a0, where a0 is the radius at which the electron is orbiting the atom.

Question 30

How do you find the minimum energy required to ensure an electron is contained within an atom?

Answer 30

Use Δp >= ħ/a0, where a0 is 10^-10, and then use KE = p^2/(m*2)

Question 31

What is an alternative expression for the uncertainy principle?

Answer 31

ΔEΔt >= ħ/2

Question 32

What do you have to do if a particle is no longer free?

Answer 32

Add a term to the TDSE for potential energy (add VΨ), and assume that V is conservative and does not change with time.