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PHYS 342 Modern Physics Course Review

Great Job !

The research projects are fantastic!

In the end (current points + final exam),  passing 60% points means at least C. 

Currently, almost everyone has more than 65% points (everyone has C+ so far).

Physics 2DOur Second Topic:

Wave-particle duality & Quantum mechanics

Our Third Topic:

Atomic physics

Nuclear physics

Particle physics

Cosmology, Statistical physics

1. Special Relativity:a) Einstein’s Postulatesb) The relativity of time and lengthc) Lorentz Transformd) Velocity Transformatione) Conservation laws in relativityf) The relativistic dynamics: momentum and energy 

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The form of each physics law is the same in all inertial frames of reference.

Light moves at the same speed relative to all frames of reference.

Einstein (1879-1955)

Einstein’s Postulates of Relativity Coordinate Transformation

L = Lo [1- (u 2/ c 2)] 1/2

Δto is the “time duration” in the object “own” frame. Δt is the “time duration” in the “lab” frame.

L0 is the “space separation” in the object “own” frame. L is the “space separation” in the “lab” frame.

Δt = Δto / [1- (u 2/ c 2)] 1/2

Δto=t’2-t’1, Lo=x’2-x’1

Δt=t2-t1, L=x2-x1

Hendrick Lorentz (1853-1928)t t’ x x’

Lorentz Transformation

2

1

2

22

1where

'''

'

cv-

zzyy

vtxxcvxtt

Conservation Laws in Relativity

Momentum Conservation:In an isolated system of particle, the total linear relativistic momentum remains constant in all inertial frames of reference.

Energy Conservation:In an isolated system of particle, the total relativistic energy remains constant in all inertial frames of reference.

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Momentum in relativity theory

p m0 v

1 v 2

c 2

Momentum

m0/ 1v2 /c2

Energy

Einstein called the m0c2 term the rest energy of the object. This means that mc2 is the total energy of the object

This is the origin of the famous E = mc2

Relation Between Relativistic Momentum and Energy

2. Particle‐Wave duality:a) “Light is absorbed”– photoelectric  effectsb) “Light is emitted”– blackbody radiationc) “Light is scattered”– the Compton effect d)  What is a photon? – wave‐particle dualitye)  De Broglie hypothesis‐ de Broglie wavesf)  Heisenberg Uncertainty Relationshipsg) Wave Packet and Probability 

The Photoelectric Effect – Photon Interpretation

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Photon Momentum

p m0 v

1 v 2

c 2

According to special relativity, when v is close to c, E ≈ pcWhen v = c, we have E = pc

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For Photon, the rest mass is zero,and the momentum is p = E/c= h/λ

The Compton effect

The wave-particle duality

EM Wave Particle

λ = h/pEM waves

For light (rest mass is zero)

p = E/c= hf/c

λd = h/pde Broglie waves

For particles(rest mass is not zero)

p m0 v

1 v 2

c 2

Heisenberg Uncertainty Relationships

It is not possible to make a simultaneous determination of the energy and the time coordinate of a particle with unlimited precision.

It is not possible to make a simultaneous determination of the position and the momentum of a particle with unlimited precision.

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Wavefunction

use  , .

probability to find a particle ∝ ,

3. Schrodinger Equation :a) Schrodinger Equation b) How to solve Schrodinger Equation 

(Following the three steps!) c) Typical examples (infinite well, finite well, barrier, step)d) Wave function at a boundary

Simplest Schrodinger Equation

m is the mass of the particle.is the potential energy of the particle.

E is the total energy of the particle .

Instead of , , , , we look at first, which is one-dimensional time-independent Schrodinger equation.

)

How to Solve Schrodinger Equation

The same procedure as you solve second‐order differentialequations: A y’’(x)+ B y(x)=01) Write down the equation.2) Find the general solution (GE) which contains somearbitrary constants.3) Find the particular solution (PE): Use the boundarycondition and the normalization condition to determine theenergy E and the constants in GE.

For free particles, U=0 and K=E.  How to solve  one‐dimensional time independent Schrodinger equation for arbitrary U(x).

2 ) 0

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How to Solve Schrodinger Equation

The same procedure as you solve second‐order differentialequations: A y’’(x)+ B y(x)=0.1) Write the equation with U(x). If U(x) has differentregimes, then we need different equations for eachregions.2) Find the general solutions (GE) of the equations. Notethat the general solutions of the differential equationshave arbitrary constants.3) Use the boundary condition and the normalizationcondition to determine the allowed values for energy E andthe constants in GE. The result is the particular solutions(PE).

Boundary Condition for Wave Function 

Discontinuous Wave

Continuous WaveDiscontinuous Slope

Continuous WaveContinuous Slope

Not allowed !

Only for the potential height on the boundary is infinite !

Any other situations!

Schrodinger Equation ‐ 1D Infinite Potential Energy Well 

U=0

U→∞ U→∞

U(x)

xⅠ Ⅱ Ⅲ

1D Infinite Potential

a a

2a

-a a

1D Finite Potential Energy Well

U=0

U=U0

U(x)

xⅠ Ⅱ Ⅲ

1D Finite Potential

a a

2a

-a a

U=U0

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axV

axorxxU

000,0

)(0

Barrier  (E<U0)

0 a x

V0

U(x)

E

I IIIII

If a particle with the energy E moving from -x to +x.

axV

axorxxU

000,0

)(0

Step  (E>U0)

0 a x

V0

U(x)

E

I IIIII

If a particle with the energy E moving from -x to +x.

0 a

V(x)

x

V0Incoming +Reflected Transmitted

A quantum particle can penetrate a potential barrier higher than its total energy. This phenomena are classical forbidden, but very common in quantum world.

Quantum Tunneling effect4. Atomic Physics:

a) Basic Properties of Atoms b) Rutherford Model and Scattering Experimentsc) Bohr Model and Line Spectrad) Hydrogen Atome) Quantum Number and Atomic Statesf) Angular Momentum in Quantum Mechanicsg) Orbital Angular Momentum and Magnetic Dipole Momenth) Spin Angular Momentum i)  Addition of Angular Momentumj) Pauli Exclusion Principle

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Roadmap for Exploring Hydrogen Atom Quantitative Calculation about Scattering Scattering by the nuclear in an atom. The path of the scattered particle is a hyperbola (?). Smaller impact parameters give large scattering angles. 

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1 1sin 8 cos 1

,

2 4 cot 2

Bohr’s Model of the Atom

“quantization condition”. 1,2. .

(Z=1 for hydrogen)

and

Bohr radius . Ground State of H

- 13.60 eV

Two Methods to Represent Atomic StatesNow we have seven quantum numbers for a certain atomic state.

( n, l , ml, s , ms, j, mj )energy Level

orbitalAM

z‐orbitalAM

spinAM

z‐spinAM

totalAM

z‐totalAM

0 , 1 , 2 , 3

energy Level

orbitalAM

z‐orbitalAM

spinAM

z‐spinAM

totalAM

z‐totalAM

# 0 , 1 , 2 , 3

# 2 2 1

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Ltot = L1 + L2

11 m

2121 tot

22 mtottot m

21 mmmtot

Addition of Angular Momentum

sjs

sj mmm

L

S

J

Addition of Spin and Orbital AM

The total angular momentum j .

12 , 1/2

Pauli Exclusion Principle

Fundamental principle -- Pauli Exclusion PrincipleAny fundamental particles with Odd/2 spin can not have the same set of quantum numbers in a quantum system.

No two electrons can have the same set of quantum number (n, l , ml, s , ms ) in a single atom.

5. Nuclear Physics:a) Nuclear Constituents b) Nuclear Size, Shape, and Densityc) Nuclear Forcesd) Nuclear Mass and Binding Energye) Quantum States in Nuclei f) Nuclear Decayg) Nuclear Reactionh) Fission and Fusion

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6. Particle Physics:a) Particle Colliderb) Standard Model of Elementary Particlesc) Field Bosons: quantization of four basic forcesd) Higgs Boson: origin of masse) Leptons and Quarks: fundamental material particlesf) Mesons and Baryons: composite material particlesg) Conservation Laws in particle physicsh) Energy and Momentum in particle collision i) Particle Physics beyond Standard Model

Particle Classification

Note that Hadrons are NOT elementary particles, and they are

composite particles made up by quarks.mesons are bosons and hadrons.

baryons are fermions and hadrons.

Standard Model Four Fundamental Forces

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How do Elementary Particles Interact ? Comprehensive Test

Wave-particle Duality

Quantum Mechanics

Atomic physics

Nuclear physics

Particle physics

Special Relativity

Wednesday, May 4, 1:00 pm to 3:00pmSame Classroom as Lecture, LD 004