Quantum Mechanics Flashcards
What is the wave equation?
d^2 y/dx^2 = 1/v^2 * d^2 y/dt^2
What can a general solution to the wave equation be written as?
y = Aexp(ikx-iωt)
What is the time-dependent Schrodinger equation?
-ħ/2m * d^2 Ψ/dx^2 = iħ * dΨ/dt, where Ψ is a wave-function
What would you get if you substitute the general solution into the TDSE?
Just differentiate it:
-ħ^2/2m(ik)^2exp(ikx-iωt) = iħ(-iω)exp(ikx-iωt)
How do you find the solution to the TDSE?
Take the real parts and set equal to zero. Is a solution if certain conditions are met.
What is the equation for energy when you combine Planck’s equation and de Broglies equation?
E = (ħ^2*k^2)/(2m) = p^2/2m = KE
What is the meaning of Ψ?
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.
What is the normalisation condition for Ψ?
integral over all relevant space of |Ψ(x,t)|^2 dx = 1
What do you get if you use the normalisation condition for a plane wave? What does this mean?
|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.
How can you find a good solution to the wavefunction?
Need to form wavepackets to represent particle - several examples of how the wave function can be combined.
What is step one when considering wave packets?
Consider Ψ1 and Ψ2, where the TDSE applies to each separately. Then Ψ = αΨ1 + βΨ2
What is the second step when considering wave packets?
- 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)
Where is the particle when considering 2 waves in opposite directions?
|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
What happens when you consider 2 plane waves travelling in the same direction but with slightly different properties?
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
What does the probability density look like for two waves travelling in the same direction?
Similar to two waves travelling in opposite direction, but is a travelling wave moving at speed vg = ω0’
What is the equation for the group velocity?
dω/dk = ω’ = group velocity
How can we find the momentum using the group velocity?
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
What is the normalisation problem?
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
How can you solve the normalisation problem?
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.
What are wave packets? What is the equation for a wave packet Ψ?
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)
What characterises the distribution of waves in wave packets?
The width of the curve when you plot a(the amplitude) against k.
What is the equation for the width of the curve a-k?
w^2 = (integral of dk |a(k)|^2 * (k-k0))/(integral of dk (a(k))^2)
What an you do if the width w is narrow (w «_space;k0)? What does this give?
ω(k) = ω0 + ω0’ Δk. Substitute this into the equation for Ψ. This gives a solution for 2 waves combined.
What is the probability density of Ψ (wave packet)?
Ψ depends of f(s) (function of x-ω0’t = s), so probability density is |Ψ|^2 = |f(s)|^2
What happens if a(k) is a Gaussian of width w? What then happens if w is large?
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.
How would you go about measuring the quantum uncertainty in a ball for example?
Use ΔpΔx >= ħ/2. Ignore the 1/2 and rearrange to find momentum uncertainty, then use mass to calculate the velocity uncertainty.
What is the top hat distribution?
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.
What is the equation for the uncertainty principle?
ΔpΔx >= ħ/2
Consider Bohr’s Hydrogen atom. What is the equation for Δp?
Δp >= ħ/a0, where a0 is the radius at which the electron is orbiting the atom.
How do you find the minimum energy required to ensure an electron is contained within an atom?
Use Δp >= ħ/a0, where a0 is 10^-10, and then use KE = p^2/(m*2)
What is an alternative expression for the uncertainy principle?
ΔEΔt >= ħ/2
What do you have to do if a particle is no longer free?
Add a term to the TDSE for potential energy (add VΨ), and assume that V is conservative and does not change with time.
