Elementary Quantum Physics - KOCW

3.1 Photons

3.2 The Electron as a Wave

3.3 Infinite Potential Well: A Confined Electron 3.4 Heisenberg’s Uncertainty Principle

3.5 Tunneling Phenomenon: Quantum Leak 3.6 Potential Box: Three Quantum Numbers 3.7 Hydrogen Atom

“The triumph of modern physics is the triumph of quantum mechanics. Even the simplest experimental observation that the resistivity of a metal depends linearly on the temperature can only be explained by quantum physics, simply because we must take the mean speed of the conduction electrons to be nearly

independent of temperature. The modern definition of voltage and ohm, adopted in January 1990 and now part of the IEEE standards, are based on the

Josephson and quantum Hall effects, both of which are quantum mechanical phenomena.”

x

E_y

B_z

x Velocity =c

Direction of Propagation y

z

The classical view of light as an electromagnetic wave. An electromagnetic wave is a travelling wave which has time varying electric and magnetic fields which are perpendicular to each other and to the direction of propagation.

Light as a Wave

“We already know that light results in interference,

diffraction, refraction, and reflection. Therefore it is no doubt that light is a wave.

Further, it has been proved that light is an electromagnetic (EM) wave.”

Electric Field :

) sin(

) ,

( x t

_o

kx t

y

= E − ω

E

1

2

: light intensity (Energy 2

flowing per unit area per second)

o o

I = c ε E 2 / : wave number

k = π λ ω = 2 πν : angular frequency

: phase velocity

c = ω νλ k =

Young’s Double Slit Experiment : Interference

P S

₁

S

₂

Photographic film showing Young's fringes

Constructive interference Destructive interference

1 2

1 2

: constructive

1 : destructive 2

S P S P n S P S P n

λ

λ

− =

 

− =  + 

 

Diffraction Patterns

X-rays with single wavelength

Powdered crystal or polycrystalline material

Scattered X-rays

Photographic film

(b)

X-rays with all wavelengths

Single crystal

Photographic film

Scattered X-rays

(a)

Diffraction patterns obtained by passing X-rays through crystals can only be explained by using ideas based on the interference of waves. (a) Diffraction of X-rays from a single crystal gives a diffraction pattern of bright spots on a photographic film. (b) Diffraction of X-rays from a

powdered crystalline material or a polycrystalline material gives a diffraction pattern of bright rings on a photographic film.

X-rays

θ

dsinθ ^dsinθ

1 2

d d

Atomic planes θ

Crystal A

B

Detector

1

2

(c)

(c) X-ray diffraction involves constructive interference of waves being "reflected" by various atomic planes in the crystal.

2 sin d θ = n λ n = 1, 2, 3,...

: Bragg’s Diffraction Condition

The Photoelectric Effect

A V

Electrons

CATHODE ANODE

Light

Evacuated quartz tube

I

The Photoelectric Effect.

“Light can behave like a stream of particles of zero rest-mass. The only way to explain a vast number of experiments is to view light as a stream of discrete

entities or energy packets called photons, each carrying a quantum of energy hν , and momentum h/λ.”

“When the cathode is illuminated with light, if the frequency ν of the light is greater than a certain critical value ν₀ ,a current flows even when the anode voltage is zero.”

“Applying a positive voltage helps to collect more of the electrons and thus increases the current, until it saturates because all the photo- emitted electrons have been collected.”

ν

1

ν

2

ν

3

I

V

(a) Photoelectric current vs. voltage when the cathode is illuminated with light of identical wavelength but different intensities (I). The saturation current is proportional to the light intensity

(b) The stopping voltage and therefore the maximum kinetic energy of the emitted electron increases with the frequency of light υ. (Note: The light intensity is not the same)

“If, on the other hand, we apply a negative voltage to the anode, we can push back the photo-emitted electrons and hence reduce the current.”

“When the negative anode voltage V is equal to V_o, which just ‘extinguishes’ the current I, we know that the potential energy

‘gained’ by the electron is just the kinetic energy ‘lost’ by the

electron.”

The Photoelectric Effect

2 0

1 2

: kinetic energy just after photoemission

e m

eV = m

υ

= KE

I₂

V

saturation

I₁ I

-V₀

The Photoelectric Effect

KE_m

−Φ

−Φ

−Φ

Cs

υ₀₃

υ₀₂

K

υ₀₁

W

υ slope =h

The effect of varying the frequency of light and the cathode material in the photoelectric experiment. The lines for the different materials have the same slope ofhbut different intercepts.

0

“The magnitude of the saturation photocurrent depends on the light intensity, whereas the KE of the emitted electron is independent of I  Only the number of electrons ejected depends on the light

intensity..”

“The KE_m of the emitted electron depends on the frequency of light.”

0

work function KE

m

= h ν − h ν

ν03

ν02 ν₀₁ ν

6.6 10 34

:Plank's constant h = × ⁻ J s⋅

“ν₀₁, ν₀₂… depend on the materials used in the experiment.

Nevertheless the slope is same.”

Work Function

0

KE

m

= h ν − h ν = h ν − Φ

“Einstein proposed that light consists of energy packets, each of which has the

magnitude hν. We can call these energy quanta photons. When one photon strikes an electron, its energy is transferred to the electron. The whole photon becomes absorbed by the electron. Yet, an electron in a metal is in a lower state of

potential energy than in vacuum.

By an amount Φ, which we call the work function of the metal..”

The lower PE in metal is a result of the Coulombic attraction interaction between the electron and the positive metal ions.

Einstein’s Theory Presented in 1905 (Nobel Prize in 1921) : Plank postulated E = nhv, n=0,1,2,3…

Idea : In the process of going from energy state nhv to (n-1)hv, the source would emit a discrete burst of electromagnetic energy hv.

That is, KE = hν

When the electron is emitted from the surface of the metal, its kinetic energy will be

Now, Φ is the minimum energy needed by an electron to pass through the metal surface and escape the attractive forces that normally bind the electron to the metal.

Einstein’s Analysis of the photoelectric experimental results :

KE

m

= h ν − Φ

h ν

0

Φ =

Millikan’s Work Presented in 1914 (Nobel Prize in 1923) :

Millikan had proved that the slope h is 6.57x10^-34 (in real 6.6262x10^-34) joule-sec.

Based on these results, we conclude that “light energy is

contained in discrete units rather than in a continuous distribution of energy. That is, light propagates as a series of little packet of

energy which later called as photon.”

c

Intuitive visualization of light consisting of a stream of photons (not to be taken too literally) [From © R. Serway, C. J. Moses and C.A. Moyer, Modern Physics, Saunders College Publishing, 1989, p.56, Fig. 2.16(b)]

x y

z

1

2

2

^{o o}

I = c ε E

I = Γ

ph

h ν A t N

∆

= ∆

Γ

_ph ^ph

Classical light intensity Light intensity Photon Flux

2 0

2

1 2

E m c

υ

c

=

−

“Can a particle exist which has no rest-mass, but which nevertheless exhibits such particle-like properties as energy (kinetic energy only) and momentum?”

Compton Scattering

0 2

1 2

P m

c

υ

=

υ

− If m₀ = 0 & v < c, then E = P = 0

If m₀ = 0 & v = c, then E & P are

indeterminate ⇒ can have any value!!!

When an X-ray (light) strikes an electron, it is deflected or “scattered (particle).” See Fig. 3.9 and apply the conservation of linear momentum.

c φ

θ

Recoiling electron

Electron X-ray photon

Scattered photon

υ,λ

y

x c

υ',λ'

'

KE

m

= h ν − h ν

p h

= λ

Compton Experiment

λ₀

IntensityofX-rays

λ θ= 0¡

Primary beam θ= 90^o

λ' λ₀

IntensityofX-rays

λ

θ= 135^o

λ₀ λ'

IntensityofX-rays

λ (b) Results from the Compton experiment

Collimator Source of

monochromatic X-rays

X-ray beam Unscattered x- rays

X-ray spectrometer

θ

Path of the spectrometer

λ'

λ₀ λ₀

(a) A schematic diagram of the Compton experiment.

Sample

Unscattered x- rays

The momentum of the photon is related to its wavelength based on the Compton experiment,

2 ,

2 2 2

E h h

p h k

ν πν ω

π π π λ

= = ⋅ =

= ⋅ =





“The photoelectric experiment and the Compton effect are just two convincing experiments in modern physics that force us to accept that light can have particle- like properties.”

How do we know whether light is going to behave like a wave or a particle ?

 We should perhaps view the two interpretations as two complementary ways of modeling the behavior of light when it interacts with matter, accepting the fact that light has a dual nature.

Black Body Radiation

I_λ

λ(µm)

1 2 3 4 5

0

3000 K

2500 K

Classical theory Planck's radiation law

Spectralirradiance

Hot body

Escaping black body radiation

Small hole acts as a black body

Schematic illustration of black body radiation and its characteristics. Spectral irradiance vs

wavelength at two temperatures (3000K is about the temperature of the incandescent tungsten

All objects emit and absorb energy in the form of radiation, and the intensity of this radiation depends on the radiation wavelength and temperature of the object. When the temperature of the object is above the temperature of its surroundings, there is a net

emission of radiation energy. The maximum amount of radiation energy that can be emitted by an object is called the black body radiation.

I_λ = Emitted radiation intensity (power per unit area) per unit wavelength.



 



  −



 



= 

1 exp

2

5

2

kT hc I hc

λ λ

π

λ

Classical physics predict : I_λ ¹ ₄ and I_λ T

∝ λ ∝

Planck’s black body radiation formula :

p h

= λ

“Just as a photon has a light wave associated with it that governs its motion, so a material particle (e.g., an

electron) has an

associated matter wave (or pilot wave) that

governs its motion.” --- A Grand Symmetry of Nature

De Broglie Relationship

“It has been found that an electron traveling with a momentum p behaves like a wave of wavelength λ given by :

Electron diffraction fringes on the screen

Filament

50kV

Electrons

Two slits

Vacuum

Young's double slit experiment with electrons involves an electron gun and two slits in a cathode ray tune (CRT) (hence in vacuum). Electrons from the filament are accelerated by a 50 kV anode voltage to produce a beam which is made to pass through the slits. The electrons then produce a visible pattern when they strike a fluoresecent screen (e.g. a TV screen) and the resulting visual pattern is photographed (pattern from C. Jšnsson, D. Brandt, S. Hirschi, Am. J. Physics, 42, Fig. 8, p. 9, 1974.

FluorescentScreen

Evidences for Electron as a Wave

Diffraction Pattern

Gold Foil Screen

Cathode Rays: Electrons

( ) ( )

34

34 3

6.63 10

6.63 10

50 10 20 /

h J s

m kg m s m

λ υ

− −

−

× ⋅

= = = ×

× ⋅

De Broglie Wavelength of many objects

A 50g golf ball traveling at a velocity of 20 m/s

Firing a stream of golf balls at a wall will not result in diffraction rings” of golf balls

A proton traveling at 2200 m/s  0.18nm is smaller than inter-atomic distance.

( ) ( )

34

10 27

6.63 10

1.8 10

1.67 10 2200 /

h J s

m kg m s m

λ υ

− −

−

× ⋅

= = = ×

× ⋅

Electron accelerated by 100V  0.123nm is smaller than inter-atomic distance.

2

100 , 1.23 10

10

2

_e

p h

eV m

m λ p

⁻

= = = ×

Schrodinger Equation

A general equation that describes the wave-like behavior of a particle as a function of appropriate potential energy and boundary conditions :

0 )

2 (

2 2

2 2

2 2

2

+ − =

∂ + ∂

∂ + ∂

∂

∂ ψ ψ ψ ψ

V m E

z y

x 

A traveling electromagnetic wave resulting from sinusoidal current oscillations, or the traveling voltage wave on a long transmission line, can generally be

described by a traveling-wave equation :

( , )

0

exp ( ) ( ) exp( )

E x t = E j kx − ω t = E x − j t ω

( ) 0 exp( ) E x = E jkx

Schrodinger Equation

P S₁

S₂

Photographic film showing Young's fringes

Constructive interference Destructive interference

Electron diffraction fringes on the screen

Filament

50kV

Electrons

Two slits

Vacuum

FluorescentScreen

“The average intensity

depend on the square of the wave amplitude. E_o² for the resultant wave depends on y axis.”

“What changes in the y direction in this case is the probability of

observing electrons.”

Schrodinger Equation

In 1926, Max Born suggested a probability wave interpretation for the wave-like behavior of the electron :

( , )

0

sin( )

E x t = E kx − ω t

^“A plane traveling wave function for an electric field, and the intensity of wave IE(x,t)I² can be measured.”

We may suggest a similar wave function for the electron : Ψ(x, t)

This assumption means that IΨ(x, t)I² represents the probability of finding the electron per unit distance.

Therefore, the wave function must have one of the following interpretations :

① IΨ(x, y, z, t)I² is the probability of finding the electron per unit volume at x, y, z, at time t.

② IΨ(x, y, z, t)I²dxdydz is the probability of finding the electron in a small elemental volume dxdydz at x, y, z at time t.

③ If we are just considering one dimension, then the wave function is Ψ(x, t), and IΨ(x, t)I²dx is the probability of finding the electron between x and (x+dx) at time t.

Schrodinger Equation

If the PE of the electron is time independent, which means that V=V(x) in one dimension, then the spatial and time dependence of Ψ(x, t) can be separated, just as in Eq. 3.9. Then

 

 

 −

=

Ψ 

x jEt t

x, ) ( )exp

( ψ

Is PE of the electron time independent ?

The net force impinging on the electron is F = -dV/dx, for example, electrostatic potential energy of the electron due to the attraction by the proton. Note V(r) is time independent.

2

0

( ) 4 V r e

πε r

= −

The fundamental equation that describes the electron’s behavior by determining Ψ(x), is called the time-independent Schrödinger equation

0 )

2 (

2 2

2 2

2 2

2

+ − =

∂ + ∂

∂ + ∂

∂

∂ ψ ψ ψ ψ

V m E

z y

x 

Schrodinger Equation

The solution of this equation gives the steady-state behavior of the electron in a time-independent potential energy environment described by V = V(x, y, z). That is, we can calculate the probability distribution and the energy of the electron :

: the steady-state probability distribution of the electron

2 2

( x, y, z,t ) ψ ( x, y, z )

Ψ =

Boundary Conditions : ψ(x) and dψ(x)/dx must be continuous.

Furthermore, ψ(x) must be single- valued.

ψ(x) not single valued

x ψ(x)

not continuous dψ

dx

x ψ(x)

x ψ(x) ψ(x) not continuous

“The time independent Schrodinger equation can be viewed as a mathematical crank. We input the

potential energy of the electron and the boundary conditions, turn the crank, and get the probability distribution and the energy of the electron under steady-state conditions.”

Example 3.5 The Free Electron

Solve the Schrödinger equation for a free electron whose energy is E. What is the uncertainty in the position of the electron and the uncertainty in the momentum of the electron?

2 2

2

2 2 2

2 0 0

d m d

E k

dx dx

ψ + ψ = → ψ + ψ =



( ) exp( )

( ) exp( )

x A jkx

x B jkx

ψ ψ

 =

 = −



Free electron  No potential energy, so the Schrodinger equation becomes:

The total wave function is obtained by multiplying ψ(x) by exp(-jEt/ħ):

( ) exp ( )

( ) exp ( )

x A j kx t

x B j kx t

ω ω

Ψ = −

 Ψ = − −



2

2

k = 2m E



Traveling waves in the +x and –x directions with k 2π

= λ

The energy E of the electron is simply KE (because of free electron), so

2 2

2 KE E k

= =  m

Example 3.5 The Free Electron

When we compare this with the classical physics expression KE = (p² / 2m), then

p =  k or p = h λ

The probability distribution for the electron is

2 2 2

( ) x A exp ( j kx ) A

Ψ = =

: constant over the entire space

※ The uncertainty Δx in its position is infinite.

※ Since the electron has a well-defined wavenumber k, its momentum p is also well-defined by virtue of p = ħk. Therefore, the uncertainty Δp in its momentum is zero.