"Plane Wave" solutions to wave equation (Term 2) Flashcards
How can we write Acos(kx)+Bsin(kx_ in another way?
= Acos(kx+phi) = Bsin(kx+phi) = Gexp(ikx)
What is another way of writing the solution to the wave equation?
u(x,t) = G1exp(i(kx-wt)) + G2exp(i(kx+wt)) (right and left moving waves)
How can we write the wave equation in 3D?
d^2u/dt^2 = c^2*((grad^2)u)
How do we write the solution to the wave equation in 3D?
Instead of kx, add ly and nz, or k*r, both vectors
How do you check if something is a solution to the wave equation?
Do the differentials in the wave equation and see if they make the wave equation.
What is the equation for a plane?
k.r = d
How is the equation for a plane useful?
Can see the wave equation with k.r in the exponent as planes moving in +ve k-direction as t increases
What is the equation for d, the position of the plane wave?
d = nλ
What 2 things does k tell us in the wave equation?
- Which direction the wave is moving in
- The waves wavelength, since k = 2π/λ
Write the time dependent Schrodinger Equation in 3D.
iħ*dΨ/dt = -ħ^2/2m ∇^2Ψ + VΨ
What can we write Ψ(r,t) as?
Ψ(r,t) = R(r)T(t) = X(x)Y(y)Z(z)T(t)
What is the new version of the Schrodinger equation with the equation for Ψ?
iħ * 1/T * dT/dt = -ħ^2/2m * 1/R * ∇^2*R + V = E (after dividing through by RT)
How do we find a solution for T(t)?
Use iħ * 1/T * dT/dt = E and multiply through by -i/ħ. Find that T(t) = G*exp(-iE/ħ * t)
What 2 cases do we consider for the Schrodinger equation?
Infinite potential well and the finite potential well.
What 2 conditions for the infinite potential well do we consider for Ψ?
Ψ(0) = Ψ(L) = 0
How do we work out X(x) for the infinite potential well?
- Assume X(x) = Acoskx + Bsinkx
- For X(0) = 0, A = 0, and for X(L) = 0, k=nπ/L
- X(x) = Bsin(nπx/L)
What is the final equation for Ψ(x,t) using all the worked out equations?
Ψ(x,t) = sum over n of Bnsin(nπx/L)exp(-iE/ħ * t) = X(x)*T(t)
What happens to the wave in the finite potential well problem?
Continues as normal in the well, and decays exponentially outside the well as it is finite.
What do we need to solve for the finite potential well problem?
The other part of the Schrodinger equation with R in it: -ħ^2/2m * 1/R * ∇^2*R + V = E
What is the first step in solving the R part of the Schrodinger equation? What do we assume?
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
What is the first step to solve 1/R * ∇^2*R = -k^2?
- Solve in 1D where ∇^2 -> d^2/dx^2, R-> X(x)
- X(x) = Acoskx + Bsinkx = A’ * exp(ikx)
What is the first step to solve 1/R * ∇^2*R = -k^2 in 3D?
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
What equation do you get with just -k^2 on one side? What do we do with this equation?
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
What is the equation for k^2?
k^2 = 2m/ħ^2 *(E-V)
What equation do we find for the finite potential well problem for X(x)? What do we do with this?
X(x) = Acoskx + Bsinkx, where k is a constant for x
For X(0), A=0, for X(L) = 0, k = nπ/L
What do we do after finding solutions for the X part?
Find solutions for the Y and Z parts too by separating them to their own sides of the equation.
What is the final solution for Ψ after finding the X, Y and Z parts?
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)
What is the equation for the energy of the particle if V=0 in the box?
E = ħ^2 / 2m *(π/L)^2 * (n^2 + m^2 + l^2)
How do you find the groundstate energy? How do you find it for higher states?
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.
