QM in 3D Flashcards
What is the 3D version of the TISE?
-ћ/2m *∇^2 u(r) + V(r)u(r) = Eu(r), where ∇^2 is the laplacian operator and r is the position vector
Which 3 ODEs do we get when we separate the TISE with central potential u(r)?
- Equation for ф(Ф) with new constant m(l)
- Equation for Θ(θ) with new constant l(l+1)
- Equation for R(r) involving the potential U(r)
What does m(l) represent? What does l represent?
m(l) = magnetic quantum number l = orbital quantum number
What is the first ODE?
d^2ф/dФ^2 + m(l)^2 *ф = 0
How can we solve the first ODE?
Try ф(Ф) = Aexp(im(l)Ф), and sub into ODE: find it works
What do we find from solving the first ODE?
That ф must be single-valued: ф(Ф) = ф(Ф+2kπ), so Aexp(im(l)2π) = 1, so m(l) = 0, +/-1, +/- 2 etc
How is l related to m(l)?
l >= |m(l)|
What do we get from combined Θ and ф solutions?
Spherical harmonics.
What do we need to solve the third ODE?
The potential U(r). Most important is coulomb potential U(r) = -e^2/4πε0r
What equation do we obtain for the enrgy En?
En = -me^4/(2(4πε0)^2*ћ^2) * 1/n^2, where n = 1,2,3… and n>l+1 (n = principle quantum number)
What can the 3 quantum numbers equal?
n = 1,2,3…, l = 0,1, … (n-1), m(l) = 0, +-1, +-2, … +-l
Both energy and angular momentum are quantised.
What happens if the energy E in the third ODE is >0?
The electron is free and the H atom is ionised.
What is the equation for the total energy E?
E = Kr + K0 - e^2/4πε0r, where Kr is purely radial motion and K0 is orbital motion
When is the square bracket in the radial equation (third OD) purely radial?
Only when K0 cancels with the l(l+1) term, so K0 must equal this term.
What is the third ODE equal to?
1/r^2 * d/dr(r^2 * dR/dr) + [2m/ћ^2 * (E-U(r)) - l(l+1)/r^2]*R = 0
What does K0 therefore have to equal?
K0 = ћ^2 *l(l+1)/2mr^2
How can we relate K0 to classical angular momentum?
L = mv0r, so K0 = 1/2mv0^2
How can we find L from K0?
ћ^2 l(l+1)/2mr^2 = 1/2mv0^2 = 1/2L^2/mr^2, where L^2 = ћ^2 *l(l+1) -> L quantized in units of ћ
What does the quantum number l mean?
l determines the magnitude of the angular momentum, but the angular momentum is a vector.
What direction does the vector L point in for a plane rotating?
Points perpendicular to the plane of rotation.
What does the quantum number m(l) mean?
Lz = ћ*m(l)
If, for example, l = 2, what does L and Lz equal?
L = ~ 2.45ћ, Lz = +-2ћ, +-ћ, 0 -> L cannot point along z-axis so Lz < |L|
What would be the problem if L pointed along the z-axis exactly?
The linear momentum p would have no z-component, so HUP would equal infinity which is not possible for a localised atomic state.
Which components of L can we know?
Can only know one component in one direction, so if we know Lz, we can’t know Lx and Ly.
Why do we ‘choose’ a z-axis?
Example of external field e.g. magnetic field with orbiting electron as current loop, magnetic moment of electron m = πr^2 * I z(hat)
-Component of L parallel to B-field is Lz which determines strength of magnetic interaction.
