1.1: Classical Physics of the 1890s

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Modern Physics for Scientists and Engineers International Edition, 4th Edition

1. THE BIRTH OF MODERN PHYSICS 2. SPECIAL THEORY OF RELATIVITY

3. THE EXPERIMENTAL BASIS OF QUANTUM PHYSICS 4. STRUCTURE OF THE ATOM

5. WAVE PROPERTIES OF MATTER AND QUANTUM MECHANICS I 6. QUANTUM MECHANICS II

7. THE HYDROGEN ATOM 8. ATOMIC PHYSICS

9. STATISTICAL PHYSICS

10. MOLECULES, LASERS, AND SOLIDS

11. SEMICONDUCTOR THEORY AND DEVICES 12. THE ATOMIC NUCLEUS

13. NUCLEAR INTERACTIONS AND APPLICATIONS 14. PARTICLE PHYSICS

15. GENERAL RELATIVITY

16. COSMOLOGY AND MODERN ASTROPHYSICS

Review:

1.1: Classical Physics of the 1890s



Mechanics



Electromagnetism



Thermodynamics

Triumph of Classical Physics:

Conservation Laws

• Conservation of energy

• Conservation of linear momentum

• Conservation of angular momentum

• Conservation of charge

• Conservation of mass ??

1.6: Unresolved Questions of 1895

2.2: The Michelson-Morley Experiment

Ether does not exist!

(no preferred frame for light)

2.3: Einstein’s Postulates

1) The principle of relativity: The laws of physics are the same in all inertial systems. There is no way to detect absolute motion, and no preferred inertial system exists.

2) The constancy of the speed of light: Observers in all inertial systems measure the same value for the speed of light in a vacuum.

The Lorentz Velocity Transformations

In addition to the previous relations, the Lorentz velocity transformations for u’_x, u’_y, and u’_zcan be obtained by switching primed and unprimed and changing v to –v:

2.5: Time Dilation and Length Contraction



Time Dilation:

Clocks (T

) in K’ run slow with respect to stationary clocks (T) in K.



Length Contraction:

Lengths (L

) in K’ are contracted with respect to the same lengths (L) stationary in K.

Consequences of the Lorentz Transformation:

Moving clocks appear to run slow

Moving objects appear contracted in the direction of the motion

0

0 2 2

1 / T T T

v c



 



2 2

0

0 1 /

L L L v c

   

Total Energy and Rest Energy

Rewriting in the form

The term mc

is called the rest energy and is denoted by E

.

This leaves the sum of the kinetic energy and rest energy to be interpreted as the total energy of the particle.

The total energy is denoted by E and is given by

2 2

K   mc  mc

E   mc 2

3.1: Discovery of the X Ray and the Electron



X rays were discovered by Wilhelm Röntgen in 1895.



Observed x rays emitted by cathode rays bombarding glass



Electrons were discovered by J. J. Thomson in 1987.



Observed that cathode rays were charged particles

 

1.76 10

/

e C kg

m  

Planck’s Radiation Law

• Quantum hypothesis makes it harder for “oscillators” at higher frequency to emit energy i.e.

eliminated the ultraviolet catastrophe.

• Still believed that the EM waves behaved classically, but that the energy transfer process is quantized.

En  nhf

E hf

 

2

5 /

2 1

( , )

hc kT

1 T c h

e

 

 

 

3.6: Photoelectric Effect

 Einstein’s Theory



Thomson – Plum Pudding

 Why? Known that negative charges can be removed from atom.

 Problem: just a random guess



Rutherford – Solar System

 Why? Scattering showed hard core.

 Problem: electrons should spiral into nucleus in ~10^-11 sec.



Bohr – fixed energy levels

 Why? Explains spectral lines.

 Problem: No reason for fixed energy levels



de Broglie – electron standing waves

 Why? Explains fixed energy levels

 Problem: still only works for Hydrogen.



Schrödinger – probability distribution

Models of the Atom

– – – –

–

+

+

+ –

4.1 Rutherford Atomic Model

 planetary model



an atom has a positively charged core (fist to use the word “nucleus”)

surrounded by the negative electrons.

The Planetary Model is Doomed.

 Radius r must decrease!

4.4: The Bohr Model of the Hydrogen Atom

r

1) “Stationary states” (orbiting electrons in atoms do not radiate energy). These states have definite total energy.

2) EM radiation emitted/absorbed when electrons make transitions between stationary states such that hf = E = E₁ − E₂

3) Classical laws of physics do not apply to transitions between stationary states.

4) The mean kinetic energy of the electron-nucleus system is quantized such that K = n h f_orb/2, equivalent to quantizing the angular momentum of the stationary states in multiples of h/2π.

Limitations of the Bohr Model

1) Works only to single-electron atoms

2) Could not account for the intensities or the fine spectral lines

3) Could not explain the binding of atoms into molecules

5.2: De Broglie Waves = Matter waves

 photons had both wave and particle properties.

 He suggested that particles with non-zero mass should have wave properties.

 De Broglie wavelength of a “matter wave”

Electrons are diffracted like waves!  Bragg law

sin

d   n 

How to describe a moving particle as a wave?



To represent a particle which is localized in space we will need a wave representation which is localized in space

 wave packet

Wave-particle duality

 Bohr pointed out that particle-like and wave-like aspects are complimentary.

 Bohr’s principle of complementarity:

 we must use either the particle description or the wave description.

 “amplitude” of the de Broglie wave is related to the probability of finding the object at x,y,z and t.

5.6: Uncertainty Principle

Heisenberg’s uncertainty principle 

p x 2

   

• “It is impossible to know the exact position and exact momentum of an object simultaneously”.

• Result is a natural consequence of the wave description of matter and not due to any experimental limitations.

E t 2

   

Probability and Wave Function

In de Broglie description the particle is represented by a wave packet, Ψ(x,y,z,t).

IΨI² is the probability density and represents the probability of finding the particle at a given location at a given time.

5.8: Particle in a Box

(1) Particle cannot have arbitrary E

(2) Confined particle cannot have zero kinetic energy.

(3) Quantization only apparent for extremely small m or L.

 Bohr – fixed energy levels

 Why? Explains spectral lines.

 Problem: No reason for fixed energy levels

 de Broglie – electron standing waves

 Why? Explains fixed energy levels

 Problem: still only works for Hydrogen.

 Schrödinger – probability distribution

+

6. Quantum Mechanics

Origin of Quantum Mechanics is credited to

Werner Heisenberg and Erwin Schrödinger.

The Schrödinger Wave Equation

The Schrödinger Wave Equation



If particles exhibit wave properties (de Broglie), there should be a wave equation for a particle



Schrödinger found an equation for the wave function using ideas from optics.



Like Newton’s laws in classical physics there is no way to derive the Schrödinger wave equation from more basic principles.

Schrodinger equation development 

new approach to physics

 Newton’s second law and Schrödinger’s wave equation are both differential equations.

 Newton’s second law can be derived from the Schrödinger wave equation, so the latter is the more fundamental.



Writing the equation in terms of operators and letting both sides of the equation operate on the wave function:



Schrödinger equation can be considered an operator equation for the total energy.

Schrödinger equation as an operator equation

Momentum operator:

Energy Operator:

Infinite Square-Well Potential 

Finite Square-Well Potential 

6.6: Simple Harmonic Oscillators

Potentials of many other physical systems can be approximated by SHM.

Diatomic molecules

1 2

( ) 2 V x  x



 



En n n

m

 

     

 The allowed energies are quantized:

 The zero point energy (E₀) is called the Heisenberg limit ₀ ¹ E  2 

6.7: Barriers and Tunneling

 Consider a particle of energy E approaching a potential barrier of height V₀ and the potential everywhere else is zero.

 Let’s consider two cases: E > V₀ and E < V₀

incident transmitted

reflected

incident transmitted

reflected

Case I: E > V

Case II: E < V

Classical: Reflection 0 Quantum: Reflection 0





Classical: Transmission 0 Quantum: Transmission 0





(Barrier) (Tunneling)

7.2: Solution of the Schrödinger Equation for Hydrogen

 Radial equation

 Angular equation

 azimuthal equation

Schrödinger equation has been separated into three ordinary second-

order differential equations, each containing only one variable.

Full wave function for electron in hydrogen atom



The radial wave function R and the spherical harmonics Y determine the probability density for the various quantum states.



The total wave function depends on n, ℓ, and m

.



7.3: Quantum Numbers

The three quantum numbers:

 n Principal quantum number

 ℓ Orbital angular momentum quantum number

 m_ℓ Magnetic quantum number

The boundary conditions:

 n = 1, 2, 3, 4, . . . Integer

 ℓ = 0, 1, 2, 3, . . . , n − 1 Integer

 m_ℓ = −ℓ, −ℓ + 1, . . . , 0, 1, . . . , ℓ − 1, ℓ Integer

The restrictions for quantum numbers:

n m r   Rn r Ym   Ym   f m  gm 

  

    

    

Orbital Angular Momentum Quantum Number ℓ



This result disagrees with the Bohr’s semiclassical “planetary” model where L = nħ (n=1,2,3,…).

 In particular, for an ℓ = 0 state the quantum result predicts:



For a given energy level there are a number of possible values for ℓ .

 Thus energy level is degenerate (energy is independent of ℓ in one electron atom).

 In many electron atom electrons with lower ℓ values have lower energy.



Ascribe letter names to the various ℓ values.

 ℓ = 0 1 2 3 4 5 6. . .

 Letter = s ^(sharp) p (principal) d (diffuse) f (fundamental) g h i. . .

( 1) L     

n = 1, ℓ = 0, L = 0, Lyman series (ultraviolet) n = 2, ℓ = 1, L = √2ħ, Balmer series (visible)

n = 3, ℓ = 2, L = √6ħ, Ritz-Paschen series (short wave infrared) n = 5, ℓ = 3, L = 2√5ħ, Pfund series (long wave infrared).

Magnetic Quantum Number m _ℓ

Relationship between quantum numbers

orbital values Number of values for m_l

s l = 0; m_l = 0 1

p l = 1; m_l = -1, 0, +1 3

d l = 2; m_l = -2, -1, 0, +1, +2 5

f l = 3; m_l = -3, -2, -1, 0, +1, +2, +3 7 g l = 4; m_l = -4, -3, -2, -1, 0, +1, +2, +3, +4 9

( 1) L     



Only magnitude and one component of the angular

momentum L can be specified simultaneously.



The spectral lines emitted by atoms in a magnetic field split into multiple energy levels.

 It is called the Zeeman effect.

 If line is split into three lines ➝ Normal Zeeman effect

 If line splits into more lines ➝ Anomalous Zeeman effect (Chapter 8)

Normal Zeeman effect 

7.4: Magnetic Effects on Atomic Spectra The Normal Zeeman Effect

: Magnetic moment of Hydrogen atom

m_ℓ Energy 1 E₀ + μ_BB

0 E₀

−1 E₀ − μ_BB



An atomic beam of particles in the ℓ = 1 state pass through a magnetic field along the z direction.



The m

= +1 state will be deflected down, the m

= −1 state up, and the

= 0 state will be undeflected.  always odd (2ℓ + 1)



But, they observed only two lines!



The atoms are deflected either up or down!  Why?

Stern-Gerlach Experiment

 Intrinsic Spin (fourth quantum number: m

)

 Electron must have an intrinsic angular momentum

Intrinsic Spin (fourth quantum number: m

)



The spinning electron reacts similarly to the orbiting electron in a

magnetic field.  Spin will have quantities analogous to L, L

, ℓ, and m

.



like L, the electron’s spin can never be spinning with its magnetic moment μ

exactly along the z axis.



The z – component of the spin angular momentum:

 The magnetic spin quantum number m_s has only two values, since (2s+1) = 2

 depending on the value of m_s , the spin will be either “up” or “down”.

 Each electronic state in the atom now described by four quantum numbers (n, ℓ, m_ℓ, m_s)

s



sz

m

L 















2 1 2 1 ms

7.6: Energy Levels and Electron Probabilities



For hydrogen, the energy level depends on the principle quantum number n.

 these energies are predicted with great accuracy by the Bohr model.

 In a magnetic field the degeneracy is removed.



In many-electron atoms the degeneracy is also removed either because of internal B fields or because the average potential of an electron due to nucleus plus

electrons is non-Coulombic.

 generally smaller ℓ states tends to lie at lower energy for a given n.

 For example, sodium

E(4s) < E(4p) < E(4d) < E(4f).

(hydrogen atom)

Selection rules for radiative transition

 In ground state an atom cannot emit radiation.

It can absorb electromagnetic radiation, or gain energy through inelastic bombardment by

particles.

 Can use the wave functions to calculate

transition probabilities for electrons changing from one state to another.

 Allowed transitions: (for photon)

 Electrons can absorb or emit photons to change states when ∆ℓ = ±1.

 Change in ℓ implies a ∆L_Z of ±ħ.

 Conservation of angular momentum means photon takes up this angular momentum.

 Forbidden transitions:(for photon)

　 Other transitions possible but occur with much smaller probabilities when ∆ℓ ≠ ±1.

1 0, 1 n anything

m

 

  



 



Selection rule

Probability Distribution Functions: Electron “cloud”

 

( , , ) , ,

n m

r   n m

 

 

