"Plane Wave" solutions to wave equation (Term 2)

Question 1

How can we write Acos(kx)+Bsin(kx_ in another way?

Answer 1

= Acos(kx+phi) = Bsin(kx+phi) = Gexp(ikx)

Question 2

What is another way of writing the solution to the wave equation?

Answer 2

u(x,t) = G1exp(i(kx-wt)) + G2exp(i(kx+wt)) (right and left moving waves)

Question 3

How can we write the wave equation in 3D?

Answer 3

d^2u/dt^2 = c^2*((grad^2)u)

Question 4

How do we write the solution to the wave equation in 3D?

Answer 4

Instead of kx, add ly and nz, or k*r, both vectors

Question 5

How do you check if something is a solution to the wave equation?

Answer 5

Do the differentials in the wave equation and see if they make the wave equation.

Question 6

What is the equation for a plane?

Answer 6

k.r = d

Question 7

How is the equation for a plane useful?

Answer 7

Can see the wave equation with k.r in the exponent as planes moving in +ve k-direction as t increases

Question 8

What is the equation for d, the position of the plane wave?

Answer 8

d = nλ

Question 9

What 2 things does k tell us in the wave equation?

Answer 9

• Which direction the wave is moving in

- The waves wavelength, since k = 2π/λ

Question 10

Write the time dependent Schrodinger Equation in 3D.

Answer 10

iħ*dΨ/dt = -ħ^2/2m ∇^2Ψ + VΨ

Question 11

What can we write Ψ(r,t) as?

Answer 11

Ψ(r,t) = R(r)T(t) = X(x)Y(y)Z(z)T(t)

Question 12

What is the new version of the Schrodinger equation with the equation for Ψ?

Answer 12

iħ * 1/T * dT/dt = -ħ^2/2m * 1/R * ∇^2*R + V = E (after dividing through by RT)

Question 13

How do we find a solution for T(t)?

Answer 13

Use iħ * 1/T * dT/dt = E and multiply through by -i/ħ. Find that T(t) = G*exp(-iE/ħ * t)

Question 14

What 2 cases do we consider for the Schrodinger equation?

Answer 14

Infinite potential well and the finite potential well.

Question 15

What 2 conditions for the infinite potential well do we consider for Ψ?

Answer 15

Ψ(0) = Ψ(L) = 0

Question 16

How do we work out X(x) for the infinite potential well?

Answer 16

• Assume X(x) = Acoskx + Bsinkx
• For X(0) = 0, A = 0, and for X(L) = 0, k=nπ/L
• X(x) = Bsin(nπx/L)

Question 17

What is the final equation for Ψ(x,t) using all the worked out equations?

Answer 17

Ψ(x,t) = sum over n of Bnsin(nπx/L)exp(-iE/ħ * t) = X(x)*T(t)

Question 18

What happens to the wave in the finite potential well problem?

Answer 18

Continues as normal in the well, and decays exponentially outside the well as it is finite.

Question 19

What do we need to solve for the finite potential well problem?

Answer 19

The other part of the Schrodinger equation with R in it: -ħ^2/2m * 1/R * ∇^2*R + V = E

Question 20

What is the first step in solving the R part of the Schrodinger equation? What do we assume?

Answer 20

Put the R’s on one side and everything else on the other: 1/R * ∇^2*R = -2m/ħ^2 (E-V)
First assume E>V as this is consistent with classical mechanics, so 1/R * ∇^2R = -k^2

Question 21

What is the first step to solve 1/R * ∇^2*R = -k^2?

Answer 21

• Solve in 1D where ∇^2 -> d^2/dx^2, R-> X(x)

- X(x) = Acoskx + Bsinkx = A’ * exp(ikx)

Question 22

What is the first step to solve 1/R * ∇^2*R = -k^2 in 3D?

Answer 22

R = X(x)Y(y)Z(z) - cube with V -> infinite outside and V -> 0 inside

Multiply out the ∇^2*R part and then put it all on once side except -k^2

Question 23

What equation do you get with just -k^2 on one side? What do we do with this equation?

Answer 23

1/X * d^2X/dx^2 + 1/Y * d^2Y/dy^2 + 1/Z * d^2Z/dz^2= -k^2

With this we put the X part on one side and set this equal to a constant -k^2, where k is a constant for x

Question 24

What is the equation for k^2?

Answer 24

k^2 = 2m/ħ^2 *(E-V)

Question 25

What equation do we find for the finite potential well problem for X(x)? What do we do with this?

Answer 25

X(x) = Acoskx + Bsinkx, where k is a constant for x

For X(0), A=0, for X(L) = 0, k = nπ/L

Question 26

What do we do after finding solutions for the X part?

Answer 26

Find solutions for the Y and Z parts too by separating them to their own sides of the equation.

Question 27

What is the final solution for Ψ after finding the X, Y and Z parts?

Answer 27

Ψ(r,t) = X(x)Y(y)Z(z) = sum over n,m,l of A(n,m,l)sin(nπx/L)sin(nπy/L)sin(nπz/L)exp(-iE/ħ * t)

Question 28

What is the equation for the energy of the particle if V=0 in the box?

Answer 28

E = ħ^2 / 2m *(π/L)^2 * (n^2 + m^2 + l^2)

Question 29

How do you find the groundstate energy? How do you find it for higher states?

Answer 29

n=m=l=1, so E = 3ħ^2 / 2m *(π/L)^2

For higher states, set one of the letters equal to 2 and keep the other 2 at 1 (can be any of the letter), so could be n=m=1, l=2, and so on.