Electronic- Quantum Mechanics Flashcards
What does the simple model of a metal explain?
Electrical and thermal conductivity because there is a sea of free electrons which bind together positive ions on a simple lattice
What does the simple model of a metal not explain?
The specific heat of a metal. The free electrons don’t significantly contribute to specific heat
Light diffraction through single slit equation
D=λl/s
D is distance from centre to first minimum
l is distance from slit to screen
s is slit width
de Broglie wavelength equation
λ=h/p
h is Planck’s constant
p is momentum
How to get diffraction pattern from electrons
Fire electrons at a crystal lattice structure and they diffract and interfere to form the diffraction pattern
What does Schroedinger’s equation describe?
The position of a quantum particle
Schroedinger’s equation
-(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
Does a free particle have boundary conditions?
No so any wavelike solution with any wavelength, momentum and energy is possible
Boundary conditions for particle in a box and solution
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…
Sommerfeld modification of free electron model
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.
Assumptions for simple model of electrons in metals as opposed to solving all the wave equations
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.
Solutions to simple model for metals
ψ(x)=asin(πnx/L)=asin(kx)
Subbing in solutions to simple model for metals into wave equation
E=h^2n^2/8mL^2
h is planck’s constant this time
Describe the wavefunction and how it is used
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)
Formula for size of energy unit in 3D
E=(h^2/8mL^2)(nx^2+ny^2+nz^2)
The 4 quantum numbers
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
Pauli exclusion principle
No two electrons can have the same four quantum numbers. Means electrons with the same set of quantum numbers in different atoms are identical.
What happens at 0K?
All atoms are in their lowest energy state - same for each atom and electron
How do Fermi-Dirac statistics work?
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.
Where is the Fermi level?
The maximum level occupied at 0K depending on number of electrons
Calculating Fermi level
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
Formula for electron number density N
N=na x ne
na is atomic number density
ne is number of conduction electrons per atom
Fermi levels compared to thermal energy, visible photon and binding energy of inner electrons
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
