Schrodinger Equation in Three Dimensions Flashcards
Schrodinger Equation in 3D
Time Dependent
iℏ ∂ψ(|r,t)/∂t= -ℏ²/2m ∇²ψ(|r,t) + V(|r,t)ψ(|r,t)
Schrodinger Equation in 3D
Probability Density
ψ(|r,t) |²
Schrodinger Equation in 3D
Normalisation
∫ | ψ(|r,t) |² dV = 1
Schrodinger Equation in 3D
Time Independent
- ℏ²/2m ∇²ψ(|r) + V(|r)ψ(|r) = E ψ(|r)
- i.e. V=V(|r) => ψ(|r,t) = ψ(|r) * e^(-itE/ℏ)
3D Infinite Potential Well
Description
- consider a box with no potential inside (V=0) and infinite potential outside (V=∞)
- solve the Schrodinger equation inside the box and then impose the condition that solutions must be zero at the walls of the box, i.e standing waves in the box
3D Infinite Potential Well
Proof
-the potential is time independent so the Schrodinger equation takes the form:
-ℏ²/2m ∂²ψ(|r)/∂x² - ℏ²/2m ∂²ψ(|r)/∂y² - ℏ²/2m ∂²ψ(|r)/∂z²
= E ψ(|r)
-different coordinates appear independently so seek solution of the form:
ψ(|r) = F(x)G(y)H(z)
-sub into the Schrodinger equation
-this gives rise to 3 equations:
d²F/dx² * 1/F + FCx = 0
=> Enx = nx²ℏ²π²/2mLx² , nx = 1,2,3,…
Fnx(x) = [2/Lx]sin(nxπx/Lx)
-and similarly for y and z
-total energy:
E = Enx + Eny + Enz
= ℏ²π²/2m * [nx²/Lx² + ny²/Ly² + nz²/Lz²]
-total wave function:
ψ(|r) = F(x)G(y)H(z)
= √[8/LxLyLz]sin(nxπx/Lx)sin(nyπy/Ly)sin(nzπ*z/Lz)
Degeneracy of the Energy Spectrum
1D
-in the 1D case:
En = ℏ²π²n²/2mL²
-for each En, there exists a unique wave function:
φ(x) = √[2/L] * sin(πnx/L)
-one wave function per energy level, no degeneracy
Degeneracy of the Energy Spectrum
3D
- different combinations of nx, ny and nz can have the same energy but different wave functions
- in the 3D case, SOME energy states are degenerate, this is due to symmetry of the well
- when the symmetry breaks, degeneracy is removed
Schrodinger Equation for a Central Potential
[^K+^U] ψ(|r) = Eψ(|r)
Hydrogen Atom
^K
^K = -ℏ²/2m [∂²/∂x² + ∂²/∂y² + ∂²/∂z²]
Hydrogen Atom
|^L²
|^L² = -ℏ²[1/sinθ ∂/∂θ(sinθ∂/∂θ) + 1/sin²θ ∂²/∂φ²]
Hydrogen Atom
Schrodinger Equation
- ℏ²/2m [1/r ∂/∂r (r² ∂/∂r) - 1/ℏ²r² |^L]*ψ(|r) + U(r)ψ(|r) = Eψ(|r)
- > where U(r) = -e²/(4πϵo*r)
Angular Momentum
Classical vs Quantum
-in classical physics: |L = |r x |p -in quantum mechanics: |^L = |^r x |^p ^Lx = ^y^pz - ^z^py ^Ly = ^z^px - ^x^pz ^Lz = ^x^py - ^y^px
Can the three quantum angular momentum operators ^Lx, ^Ly, ^Lz be measured simultaneously?
-operators can be measure simultaneously IF [^A,^B]=0
-but:
[^Lx , ^Ly] = -iℏ^Lz
[^Ly , ^Lz] = -iℏ^Lx
[^Lz , ^Lx] = -iℏ^Ly
-therefore we cannot measure all three components of angular momentum at the same time
Eigenvalue Problem of ^|L
-define ^|L² = Lx² + Ly² + Lz²
=>
[^Lz , ^|L²] = 0
[^Ly , ^|L²] = 0
[^Lx , ^|L²] = 0
-this means we can pick one component e.g. ^Lz and diagonalise it together with ^|L²
-the common eigenvectors of these two operators are:
{ |l*ml⟩ }
-corresponding eigenvalues:
^|L² |l*ml⟩ = ℏ²l(l+1) |l*ml⟩
^Lz |l*ml⟩ = ℏm |l*ml⟩
-therefore we need two numbers to label the eigenvalues and eigenvectors of these operators, l and mlPossible Values of l and ml
l = orbital quantum number, must be an integer
ml = azimuthal quantum number takes values in the interval -l to l inclusive
e.g. l=0 => ml = 0
l=1 => ml = -1,0,1
l=2 => ml = -2,-1,0,+1,+2
etc.
Spherical Harmonics
-in coordinate representation, the eigenvectors of angular momentum are known as spherical harmonics:
⟨ θ , φ|l*ml⟩ ⟩= Ylml(θ,φ)
-the spherical harmonics also solve the eigenvalue problem Av=λv, v = Ylml
-and they are orthonormal
What is ml graphically?
-ml is equal to the number of nodes in the interval 0 to 2π of φ
The Radial Equation
-starting with the Schrodinger equation for the Hydrogen atom, assume a solution of the form:
ψ(|r) = R(r)*Ylml(θ,φ)
-sub in and cancel Y terms to arrive at the radial equation;
-ℏ²/2m 1/r d/dr (r² dR(r)/dr) - ℏ²l(l+1)/2mr² R(r) + U(r)R(r) = ER(r)
Solutions of the Radial Equation
-allowed energies are quantised in terms of radial quantum number, n:
En = - C 1/n²
-where C is a constant, C = me^4/2(4πϵo)²ℏ²
What are the allowed quantum numbers for the electron inside the hydrogen atom?
n = 1, 2, 3, .... l = 0, 1, 2, ... , n-1 ml = -l, .... , +l
Quantum Number
Definition
-an eigenvalue of some operator ^O which commutes with ^H, [^H,^O]=0, i.e. they will not change as time goes on so can always be used to specify the state of the system
l and shells
l=0 => l=s l=1 => l=p l=2 => l=d l=3 => l=f ...
Hydrogen Wave Functions
-the total wave function of the hydrogen atom:
ψ(|r) = Rnl(r) Ylml(θ,φ)
-radial wave functions are found by solving the radial equation, and they are normalised by:
∫ r² |Rnl(r)|² dr = 1
-where the integral is from 0 to infinity
Hydrogen Atom
Level Degeneracy
-similar to the 3D potential well, for the hydrogen atom we also find degeneracy of certain levels:
-total degeneracy of level n is:
gn = n²
Hydrogen Atom
Electron’s Whereabouts
- probability density is a cloud, i.e the electron is smeared out over some region of space
- as n increases, the probability extends further from the nucleus
- in s states (l=0) probability is spherically symmetric
- in p states (l=1) one possible orbit is along the z-axis and 2 in the x-y plane
- in all state l>0, probability density vanishes at the origin
Radial Probability
-when the atom is isolated from external fields, it has spherical symmetry
-the probability of finding an electron doesn’t depend on angle but does depend on radial distance:
probability of finding electron at distance r
= r² |Rnl(t)|²
