Electronic- Quantum Mechanics

Question 1

What does the simple model of a metal explain?

Answer 1

Electrical and thermal conductivity because there is a sea of free electrons which bind together positive ions on a simple lattice

Question 2

What does the simple model of a metal not explain?

Answer 2

The specific heat of a metal. The free electrons don’t significantly contribute to specific heat

Question 3

Light diffraction through single slit equation

Answer 3

D=λl/s
D is distance from centre to first minimum
l is distance from slit to screen
s is slit width

Question 4

de Broglie wavelength equation

Answer 4

λ=h/p
h is Planck’s constant
p is momentum

Question 5

How to get diffraction pattern from electrons

Answer 5

Fire electrons at a crystal lattice structure and they diffract and interfere to form the diffraction pattern

Question 6

What does Schroedinger’s equation describe?

Answer 6

The position of a quantum particle

Question 7

Schroedinger’s equation

Answer 7

-(hbar/2m)(d2ψ/dx2)+Vψ=Eψ
E is energy of particle
V is potential energy
m is mass of particle
ψ is wavefunction of particle
Moving in x direction
h bar is h/2π

Question 8

Does a free particle have boundary conditions?

Answer 8

No so any wavelike solution with any wavelength, momentum and energy is possible

Question 9

Boundary conditions for particle in a box and solution

Answer 9

Potential inside box is 0 and outside in infinite. Means solution of form sin(kx). Must have sin(kL)=0. Means k=nπ/L where n is 1, 2, 3…

Question 10

Sommerfeld modification of free electron model

Answer 10

Assume potential energy constant inside metal so electrons don’t have a preferred location. Assume infinite potential barrier at edge of metal. Origin of barrier is positive charge of nuclei which attract an escaping electron back to the box. Means electron wave function tends to 0 outside the box. Assume electrons behave independently. Wave functions are solutions of simple wave equation.

Question 11

Assumptions for simple model of electrons in metals as opposed to solving all the wave equations

Answer 11

One dimensional, all but conduction electrons tightly bound to nuclei. 1, 2 or 3 conduction electrons freely flow through lattice independent of other electrons and atomic nuclei.

Question 12

Solutions to simple model for metals

Answer 12

ψ(x)=asin(πnx/L)=asin(kx)

Question 13

Subbing in solutions to simple model for metals into wave equation

Answer 13

E=h^2n^2/8mL^2

h is planck’s constant this time

Question 14

Describe the wavefunction and how it is used

Answer 14

It’s sinusoidal with the number of nodes increased by one for each successive state. The wavefunction squared gives the probability that the electron is at some point x. Solutions of the wave equations are possible only for certain energy values. Corresponding solutions are electron wave equations which tell us where the electron is most likely to be (not how it moves)

Question 15

Formula for size of energy unit in 3D

Answer 15

E=(h^2/8mL^2)(nx^2+ny^2+nz^2)

Question 16

The 4 quantum numbers

Answer 16

n principle quantum number: any integer
l orbital angular momentum quantum number: integer 0 to n-1
m subscript l magnetic quantum number: integer -l to l
s spin quantum number: +/- 1/2

Question 17

Pauli exclusion principle

Answer 17

No two electrons can have the same four quantum numbers. Means electrons with the same set of quantum numbers in different atoms are identical.

Question 18

What happens at 0K?

Answer 18

All atoms are in their lowest energy state - same for each atom and electron

Question 19

How do Fermi-Dirac statistics work?

Answer 19

For the electron energy levels in a metal, levels available for conduction electrons are associated with the whole specimen not just one atom. Apply PEP. Once one state is filled no more electrons can enter it. Fill boxes at increasingly higher energy levels until all electrons are in place (at 0K). No thermal energy considered and 2 electrons per box.

Question 20

Where is the Fermi level?

Answer 20

The maximum level occupied at 0K depending on number of electrons

Question 21

Calculating Fermi level

Answer 21

Consider 3D box edge L with N electrons per unit volume. Total electrons to fit in energy states is NL^3. Use formula for size of energy unit in 3D. Number of states with E

Question 22

Formula for electron number density N

Answer 22

N=na x ne
na is atomic number density
ne is number of conduction electrons per atom

Question 23

Fermi levels compared to thermal energy, visible photon and binding energy of inner electrons

Answer 23

Fermi level: 2-10ev
At 300K thermal energy: 0.04ev
Visible photon energy wavelength 550nm: 2.25ev
Binding energy inner electrons: 1000s ev
Means temperature only disturbs a few electrons near top of energy distribution