Physics Feud

Equation of Traveling wave of light

E_y(x,t) = E₀ exp(ikx − iωt)

Wave Nature of Light

• Electromagnetic wave with certain frequency

• Explains interference, diffraction, refraction, and reflection

• electric and magnetic fields are perpendicular to each other and the direction of propagation

Intensity of light wave

What does classical EM believe intensity only relies on (and is incorrect)?

Depends only on strength, not the frequency of light

X-ray DIffraction

Uses constructive interference of waves being reflected by periodic atomic structure

Why are x-rays used for diffraction?

X-ray wavelength is comparable to the spacing between atomic planes

Young’s Double Slit Experiment

• Demonstrates wave nature of light

• experiment shows bright and dark fringes due to the constructive and destructive interference

Photoelectric effect

• electrons are emitted from a metal surface when light of a high enough frequency shines on it

• Demonstrates particle nature of light

Work function

• energy needed to free an electron for metal surface because PE of electron is lower than outside energy

• varies per each element

Critical frequency

Frequency needed for the electrons to have enough energy to leave the metal surface

What happens when you apply a positive voltage to a photocathode

• Electrons are accelerated towards the anode

• Current increases until saturation

What happens when you apply a negative voltage to a photocathode

• Reduces the current

• Current becomes zero at the stopping voltage

Stopping Voltage

• minimum voltage to stop all electrons

• increases with increasing frequency

How would the current voltage plot of the photoelectric effect change according to classical theory?

The I-V curve would shift up, and the stopping voltage magnitude is greater

How does the I-V curve oft he photoelectric effect change when we change the intensify of light, but keep the frequency the same

• saturation current increases with increases in intensity

• Stopping voltage remains the same because the KE does not change

How does the I-V curve oft he photoelectric effect change when we change the frequency

• Magnitude of Stopping voltage increases with increasing frequency

• Max KE of electrons increases with increasing frequency

Main Quantum takeaways from photoelectric effect

• Higher light intensity shall lead to higher electron kinetic energy, thus higher stopping voltage V₀

• Photoelectric current and stopping voltage shall not depend on light frequency f

Quantum picture of light

Stream of energy packets that contain a frequency and momentum

Wave like behavior of light examples

• Young’s double slit experiment

• XRD

Particle like behavior of light examples

• Photoelectric effect

• Compton Scattering

• Blackbody radiation

Young’s double slit experiment of electrons

• electrons beam is fired through slits and produce a visible pattern striking a fluorescent screen

• Shows wave behavior of electrons

De Broglie Relationship

relates the wavelength of electron to its momentums (branches wave and particle behavior)

Electron as a wave examples

• Young’s double-slit experiment for electrons

• Electron diffraction

• Discrete emission and absorption spectra of hydrogen atom

Traveling wave general solution

u(x,t) = u₀ exp(ikx − iωt)

Standing wave general solution (ends are fixed, has some reflection)

u(x,t) = 2u₀exp(−iωt)cos(kx)

Probability of finding electron at a certain space and time

Total probability of finding electron in whole space at any time

1

Time independent Schrodinger’s eq

Note that:

1. V and ψ are space-dependent function

2. E is energy of electron - a space-independent scalar value.

3. For a given V(x), we can solve Schrödinger equation and obtain ψ(x) and E, thus probability density

Time independent Schrodinger’s eq in 1D

Constraints on ψ(x)

• Must be continuous

• Derivative of ψ(x) must be continuous

• must be single valued

Expectation value in quantum mechanics

Infinite potential well

• PE outside the well is infinite

• PE of electron in well is 0, but always has a finite KE

• Confined electron has quantized energy states that depend on well width and depth

• wavefunction has (n-1) nodes in the nth eigen-function (more oscillation, higher energy)

• Probability at nodes is 0

• Transition between 2 adjacent lever is also discretized (∆E = E_n+1 − E_n)

• ∆E approaches 0 as a goes to infinity (free electron)

Boundary conditions for infinite potential well

ψ(x = 0) = 0

ψ(x = a) = 0

Steps to solve for energy in infinite potential well

1. Apply general solution: ψ(x) = Ae^+jkx + Be^−jkx

2. Apply boundary conditions (boundaries are 0)

3. Convert A(e^jka -e-^jka) = 0 to sin(ka) = 0

4. Apply ka = nπ

5. Plug (2m_eE)^1/2a/ħ for k and solve E (E = (h²n²/8m_ea²))

Steps to solve for wavefunction in infinite potential well

1. Apply general solution: ψ(x) = Ae^+jkx + Be^−jkx

2. Apply boundary conditions (boundaries are 0)

3. Convert A(e^jka -e-^jka) = 0 to sin(ka) = 0

4. Use the normalization condition to set 1= integral of (2Asin(kx))² from 0 to a

5. Use identity integral of sin²(ax) = (x/2) - (sin(2ax)/4a)

6. Solve for A = 1/(2a)^1/2

Eigenenergies in infinite potential well

wave function in infinite potential well

electron probability function of infinite potential well

Expectation value of PE in infinite potential well

0

Expectation value of KE in infinite potential well

same as the eigenenergy

Electron in a finite well

• Has a finite PE

• electron has prob of penetrating in the barrier

Equation for regions 1 and 3 in finite potential well

Equation for region 2 in finite potential well

Boundary conditions for finite potential well

function and derivatives are continuous at boundaries

Steps to solve finite potential well

1. Find general functions for Regions 1 and 3: ψ_I(x) = A₁e^αx and ψ_III(x) = C₂e^−αx = C₃e^−α(x−a)

2. Reduce trial function of region 2 (ψ_II(x) = B₁e^jkx + B2e^−jkx) into symmetric and antisymmetric wavefunction: ψ_II^as(x) = B₁′ sink(x − a/2) and ψ_II^s(x) = B₂′ cosk(x − a/2)

3. Apply boundary conditions and solve numerically to get: ψ_II^as(x) : α = −k cot(ka/2) and ψ_II^s(x) : α = +k cot(ka/2)

4. Intercept of α and k are the eigenenergies

Alpha for a symmetric wavefunction in finite potential well

+k cot(ka/2)

Alpha for an antisymmetric wavefunction in finite potential well

-k cot(ka/2)

How do the eigenenergies change if the finite quantum well width increases?

decreases: increasing a squeezes k in cot function which decreases where k intercepts alpha

How do the eigenenergies change if the finite quantum well depth increases?

increases: increases outside potential which increases alphas, then increases k thus E

Finite potential well vs infinite potential well

• Finite has a finite number of energy states, while infinite has infinite

• Energy levels in finite potential well are lower because uncertainty is greater due to wall penetration, decreasing the energy

Principle Quantum Number (n)

• Quantizes electron energy

• Determines size/shell of orbital

• n = 1,2,3,…

Orbital angular momentum quantum number

• Quantizes the magnitude of orbital angular momentum

• Determines orbital shape (s,p,d,f)

• l = 0, 1, 2, ... (n − 1)

Magnetic Quantum number

• Quantizes the orbital angular momentum component along a magnetic field B_z

• Determines the orientation of orbital (p_x, p_y, p_z)

• m_l = 0, ±1, ±2, ... , ±l

Spin magnetic Quantum number

• Quantizes the spin angular momentum component along a magnetic field B_z

• m_s = ±1⁄2 (spin up vs spin down)

Hydrogenic atom model

• Energy is quantized and determines by principle quantum number

• Distance between energy levels decreases at higher and higher n

• ψ_n,l,ml(r, θ, φ) = R_n,l(r)Y_l,ml(θ, φ)

• Radial function demonstrates nodes between subshells

Quantum tunneling

• electron can leak through potential energy barrier

• In region I, the incident and reflected waves interfere to give ψ_I(x)

• In region II, the wave function tunnels through potential barrier and decays with x because E < Vo.

• In region III, no reflected wave.

Schrodinger’s eq and general wavefunction for region 1 of quantum tunneling

Schrodinger’s eq and general wavefunction for region 2 of quantum tunneling

Schrodinger’s eq and general wavefunction for region 3 of quantum tunneling

Probability of tunneling

Probability of reflecting

Probability of tunneling for a wide or high barrier

exponentially decreases with wider or higher barriers

Scanning tunneling microscopy (STM)

• Uses the tunneling effect to map atomic scale topology

• Applies a voltage to lower PE barrier between tip and sample, creating a tunneling current

Pauli exclusion principle

• In a given atom, no two electrons can have the same set of quantum numbers

• max of 2 electrons allowed in an orbital, must pair their spins (± 1/2)

Boltzmann Distribution

• Classical stats

• can describe quantum particles when there are many more available states than particles (E-E_f » kt)

• Prob at low energy levels can go to infinity

Deriving Boltzmann stats

1. Set forward and reverse rxn equal: P(E₁)P(E₂)= P(E₃)P(E₄)

2. Assume E₂ = δE, E₄ = 0: P(E₁)P(δE) = P(E₁ + δE)P(0)

3. Taylor series expansion that P(δE) ≈ P(0) + P′(0)δE & P(E₁+ δE) ≈ P(E₁) + P′(E₁)δE

4. Plug expansion in and cancel out δE: P(E₁)P′(0)δE = P′(E₁)P(0)δE

5. Use P’(E)/P(E) = P’(0)/P(0) = constant, to get general solution: P(E) = Aexp(− βE)

6. P(E) = Aexp(-E/kT)

Fermi-Dirac Statistics

• Quantum particles with half integer spin (electrons)

• Obeys pauli exclusion principle

• Prob at low energy levels goes to 1

Fermi energy

• electron occupancy probability is 50%

When can the fermi Dirac distribution be approximated by the Boltzmann distribution?

As the Energy increases ((E-E_F»kT)

How does the Fermi Dirac distribution increase at higher temps?

Curves near fermi level become more gradual

Bose-Einstein Statistics

• Quantum stats for particles with integer spin (photons, phonons)

• unlimited number of particles per state, at low energy levels, can reach infinity

Crystal

repeated, periodic, and infinite array of identical groups of atoms

Basis

single repetitive group of atoms

Lattice

the sets of the periodic mathematical points which the basis is attached to

Translation Vectors

vectors a1, a2 (a3), such that the arrangement of atoms in the crystals looks the same when the entire lattice is translated

Primitive vectors

Translation vectors that result in the smallest volume of the parallelogram/parallelopiped

Primitive Cell

• Parallelogram defined by the primitive vectors

• All primitive have the same volume

Unit cell

• contains all the essential information about the crystal and the entire crystal structures are repetitions of the unit cell.

• A primitive cell isa unit cell, but a unit cell can be non-primitive

Wigner-Seitz Cell

a primitive cell and represents the highest level of symmetry

Wigner-Seitz Cell Construction

1. draw lines to connect a given lattice point to all nearby lattice points;

2. at the midpoint and normal to these lines, draw new lines or planes.

3. The smallest volume enclosed in this way is the Wigner-Seitz cell.

How many possible rotation operations are possible for a 2D lattice

5: 1-fold (360°), 2-fold (180°), 3-fold (120°), 4-fold (90°), 6-fold (60°)

General Oblique lattice

a₁ ≠ a₂, no restriction on θ

Hexagonal Lattice

θ = 60°, a₁ = a₂

6-fold rotation, reflections

Square Lattice

θ = 90°, a1 = a2

4-fold rotation, reflections

Rectangular Lattice

θ = 90°, a1 ≠ a2

2-fold rotation, reflections

Centered Rectangular

θ = 90°, a1 ≠ a2

2-fold rotation, reflections

Why is the centered square not a special type?

a centered square lattice can just be reduced to a normal square lattice

triclinic

a ≠ b ≠ c

α ≠ β ≠ γ ≠ 90

monoclinic

a = b ≠ c

α = γ = 90, β ≠ 90

rhombohedral

a = b = c

α = β = γ ≠ 90

hexagonal

a = b ≠ c

α = β = 90, γ ≠ 120

orthorhombic

a ≠ b ≠ c

α = β = γ = 90

tetragonal

a = b ≠ c

α = β = γ = 90

Cubic

a = b = c

α = β = γ = 90

Primitive vectors and cell for simple cubic

(a,0,0), (0,a,0), (0,0,a)

Number of atoms per cell and atom positions for simple cubic

1
(0,0,0)

Number of nearest neighbor of simple cubic

6

Primitive vectors and cell for bcc

(a/2,-a/2,a/2)

(a/2,a/2,-a/2)

(-a/2,a/2,a/2)

Number of atoms per cell and atom positions for bcc

2
(0,0,0)

(a/2,a/2,a/2)

Number of nearest neighbor of bcc

8