A wave packet in a square well (an electron in a box) changing with time.

Electron in a Box

The probability of detecting a photon within a narrow region of width δx at position x is directly proportional to the square of the light wave amplitude function at that point.

Last Time: Light

2Prob(in x at x) A(x) xδ δ∝

Wave model: Interference pattern is in terms of wave intensity Photon model: Interference in terms of probability

Probability Density Function: 2( ) A(x)P x ∝

The probability density function is independent of the width, δx , and depends only on x. SI units are m-1.

There is no electron wave so we assume an analogy to the electric wave and call it the wave function, psi, : The intensity at a point on the screen is proportional to the square of the wave function at that point.

( )xΨ

The Probability Density Function is the “Reality”!

A light analogy…..

2( ) ( )P x x= Ψ

Double Slit: Electrons

The probability of detecting an electron within a narrow region of width δx at position x is directly proportional to the square of the wave function at that point:

Probability: Electrons

2Prob(in x at x) (x) xδ δ= Ψ

Probability Density Function:

2( ) (x)P x = ΨThe probability density function is independent of the width, δx , and depends only on x. SI units are m-1. Note: The above is an equality, not a proportionality as with photons. This is because we are defining psi this way. Also note, P(x) is unique but psi in not since –psi is also a solution. DEGENERACY.

2Prob( ) ( ) ( )R R

L L

x x

L Rx x

x x x P x dx x dxψ≤ ≤ = =∫ ∫

2 ( ) ( ) 1P x dx x dxψ∞ ∞

−∞ −∞

= =∫ ∫

Normalization

If the strip isn’t narrow, then we integrate the probability density function so that the probability that an electron lands somewhere between xL and xR is:

Most Probable: ( ) 0dP x

dx=

2 , 1, 2,3.....nL nn

λ = =Waves:

De Broglie: hp

λ =

Combine:

221

2 2pE mvm

→ = =

2 2

28nh nEmL

= Energy is Quantized!

Electron Waves leads to Quantum Theory

Wave Packet: Making Particles

out of Waves

Superposition of waves to make a defined wave packet. The more waves used of different frequencies, the more localized.

However, the more frequencies used, the less the momentum is known.

/p hf c=hpλ

= c fλ=

You make a wave packet by wave superposition and interference. The more waves you use, the more defined your packet and the more defined the position of the particle. However, the more waves you use of different frequencies (energy or momentum) to specify the position, the less you specify the momentum!

Heisenberg Uncertainty Principle

/ 4x p h π∆ ∆ >/ 4E t h π∆ ∆ >

Little h bar!

Which of these particles, A or B, can you locate more precisely?

A. A B. B C. Both can be located with same precision.

A. A B. B C. Both can be located with same precision.

Which of these particles, A or B, can you locate more precisely?

Heisenberg Microscope

Small wavelength (gamma) of light must be used to find the electron because it is too small. But small wavelength means high energy. That energy is transferred to the electron in an unpredictable way and the motion (momentum) becomes uncertain. If you use long wavelength light (infrared), the motion is not as disturbed but the position is uncertain because the wavelength is too long to see the electron. This results in the Uncertainty Principle.

~ /

xp h

λλ

∆∆ =

x p h∆ ∆ >

Electron in a Box

~ ~ / /p p h h Lλ∆ =

~ /x p L h L h∆ ∆ ⋅ >

~ x L∆

If you squeeze the walls to decrease ∆x, you increase ∆p!

The possible wavelengths for an electron in a box of length L.

Improved technology will not save

us from Quantum Uncertainty! Quantum Uncertainty comes from the particle-wave nature of matter

and the mathematics (wave functions)

used to describe them!

Heisenberg Uncertainty Trying to see what slit an electron goes

through destroys the interference pattern.

Electrons act like waves going through the slits but arrive at the detector like a particle.

Which Hole Did the Electron Go Through?

Conclusions: • Trying to detect the electron, destroys the interference pattern. • The electron and apparatus are in a quantum superposition of states. • There is no objective reality.

If you make a very dim beam of electrons you can essentially send one electron at a time. If you try to set up a way to detect which hole it goes through you destroy the wave interference pattern.

Feynman’s version of the Uncertainty Principle

It is impossible to design an apparatus to determine which hole the electron passes through, that will not at the same time disturb the electrons enough to destroy the interference pattern.

General Principles

Where do the Wave Functions come from???

2 2

22d ψ Uψ Eψ

m dx− + =

Solutions to the time-independent Schrödinger equation:

( )−=

2

2 2

2m U Ed ψ ψdx

OR

Where does that come from???

The Schrödinger Equation Consider an atomic particle with mass m and mechanical energy E in an environment characterized by a potential energy function U(x). The Schrödinger equation for the particle’s wave function is

Conditions the wave function must obey are 1. ψ(x) is a continuous function. 2. ψ(x) = 0 if x is in a region where it is physically

impossible for the particle to be. 3. ψ(x) → 0 as x → +∞ and x → −∞. 4. ψ(x) is a normalized function.

Wavefunction Fun

2Prob( ) ( ) ( )R R

L L

x x

L Rx x

x x x P x dx x dxψ≤ ≤ = =∫ ∫

2 ( ) ( ) 1P x dx x dxψ∞ ∞

−∞ −∞

= =∫ ∫Normalization

Most Probable: ( ) 0dP xdx

=

Probability:

Expectation “average” value:

Wave Function: Probability Amplitude: ( )xψ

2( ) ( ) P x xψ=Probability Density:

2 x = ( ) ( )xP x dx x x dxψ∞ ∞

−∞ −∞

⟨ ⟩ =∫ ∫

Quantum Cases

• Free Particle • Particle in a rigid box • Particle in a finite box “quantum well’ • Quantum Tunneling • Harmonic Oscillator • Hydrogen Atom (Chapter 42)

Wave Function of a Free Particle • The wave function of a free particle moving along

the x-axis can be written as ψ(x) = Aei(kx-ωt) – A is the constant amplitude – k = 2π/λ is the angular wave number of the wave

representing the particle – A free particle must have a sinusoidal wavefunction

because it is not confined. • Although the wave function is often associated

with the particle, it is more properly determined by the particle and its interaction with its environment – Think of the system wave function instead of the

particle wave function

Free Particle Problem

A free electron has a wave function at t=0 where x is in meters.

(a) Show that it satisfies the SE. (b) Find its de Broglie wavelength (c) Find its momentum

( ) ( )xiAex1010 00.5 ×=ψ

A Particle in a Rigid Box Consider a particle of mass m confined in a rigid, one-dimensional box. The boundaries of the box are at x = 0 and x = L.

1. The particle can move freely between 0 and L at constant speed and thus with constant kinetic energy.

2. No matter how much kinetic energy the particle has, its turning points are at x = 0 and x = L.

3. The regions x < 0 and x > L are forbidden. The particle cannot leave the box.

A potential-energy function that describes the particle in this situation is

A Particle in a Rigid Box The solutions to the Schrödinger equation for a particle in a rigid box are

For a particle in a box, these energies are the only values of E for which there are physically meaningful solutions to the Schrödinger equation. The particle’s energy is quantized.

A Particle in a Rigid Box The normalization condition, which we found in Chapter 40, is

This condition determines the constants A:

The normalized wave function for the particle in quantum state n is

A Particle in a Rigid Box

Schrödinger Equation Applied to a Particle in a Box

• Solving for the allowed energies gives

• The allowed wave functions are given by

– The second expression is the normalized wave function – These match the original results for the particle in a box

22

28nhE nmL

=

2( ) sin sinnnπx nπxψ x A

L L L = =

HW P.17

Graphical Representations for a Particle in a Box

Energy of a Particle in a Box

• We chose the potential energy of the particle to be zero inside the box

• Therefore, the energy of the particle is just its kinetic energy

• The energy of the particle is quantized

22

2 1 2 38

, , ,nhE n nmL

= =

The Quantum Jump

Quantum Energy States 2 2

28nh nEmL

=

Energy is Quantized! Only discrete energy states are allowed.

Where is the electron between jumps?

EXAMPLE :Energy Levels and Quantum jumps

QUESTIONS:

The wave function in the classically forbidden region of a finite potential well is

The wave function oscillates until it reaches the classical turning point at x = L, then it decays exponentially within the classically forbidden region. A similar analysis can be done for x ≤ 0. We can define a parameter η defined as the distance into the classically forbidden region at which the wave function has decreased to e–1 or 0.37 times its value at the edge:

Finite Potential Wells

Finite Potential Wells The quantum-mechanical solution for a particle in a finite potential well has some important properties:

• The particle’s energy is quantized. • There are only a finite number of bound states. There

are no stationary states with E > U0 because such a particle would not remain in the well.

• The wave functions are qualitatively similar to those of a particle in a rigid box, but the energies are somewhat lower because the wave functions are spread out which means lower kinetic energy.

• The wave functions extend into the classically forbidden regions.

Penetration distance of an electron

Quantum-Mechanical Tunneling

The probability that a particle striking the barrier from the left will emerge on the right is found to be

Applications of Tunneling

• Alpha decay – In order for the alpha particle to escape from

the nucleus, it must penetrate a barrier whose energy is several times greater than the energy of the nucleus-alpha particle system

• Nuclear fusion – Protons can tunnel through the barrier caused

by their mutual electrostatic repulsion

More Applications of Tunneling – Scanning Tunneling Microscope

• An electrically conducting probe with a very sharp edge is brought near the surface to be studied

• The empty space between the tip and the surface represents the “barrier”

• The tip and the surface are two walls of the “potential well”

Scanning Tunneling Microscope

• The STM allows highly detailed images of surfaces with resolutions comparable to the size of a single atom

• At right is the surface of graphite “viewed” with the STM

Quantum Tunneling Cosmology BIG BANG!

The Universe Tunneled in from Nothing

The potential-energy function of a harmonic oscillator:

where we’ll assume the equilibrium position is xe = 0. The Schrödinger equation for a quantum harmonic oscillator is then

The Quantum Harmonic Oscillator

The Quantum Harmonic Oscillator

The wave functions of the first three states are

Where ω = (k/m)–½ is the classical angular frequency, and n is the quantum number, b is classical turning point.

Energy Level Diagrams – Simple Harmonic Oscillator

• The separation between adjacent levels are equal and equal to ∆E = ω

• The energy levels are equally spaced

• The state n = 0 corresponds to the ground state – The energy is Eo = ½ ω

• Agrees with Planck’s original equations!!

Light emission by an oscillating electron

Light emission by an oscillating electron

Molecular Vibrations Molecular bonds are modeled as quantum harmonic oscillators energies below the dissociation energy.

Casimir Effect: Model the Quantum Vacuum as a harmonic oscillator. The Zero Point Energy adds up! Eo = ½ hω

The Correspondence Principle • Niels Bohr put forward the idea that the average behavior

of a quantum system should begin to look like the classical solution in the limit that the quantum number becomes very large—that is, as n → ∞.

• Because the radius of the Bohr hydrogen atom is r = n2aB, the atom becomes a macroscopic object as n becomes very large.

• Bohr’s idea, that the quantum world should blend smoothly into the classical world for high quantum numbers, is today known as the correspondence principle.

As n gets even bigger and the number of oscillations increases, the probability of finding the particle in an interval δx will be the same for both the quantum and the classical particles as long as δx is large enough to include several oscillations of the wave function. This is in agreement with Bohr’s correspondence principle.

The Correspondence Principle

I think I can safely say that nobody understands

quantum mechanics.

Richard Feynman

Copenhagen Interpretation of the Wave Function

• Quantum mechanics is a model of the microscopic world. Like all models, it is created by people for people. It divides the world into two parts, commonly called the system and the observer. The system is the part of the world that is being modeled. The rest of the world is the observer. An interaction between the observer and the system is called a measurement. Properties of the system that can be measured are called observables. The initial information the observer has about the system comes from a set of measurements. The state of the system represents this information, which can be cast into different mathematical forms. It is often represented in terms of a wave function. Quantum mechanics predicts how the state of the system evolves and therefore how the information the observer has about the system evolves with time. Some information is retained, and some is lost. The evolution of the state is deterministic. Measurements at a later time provide new information, and therefore the state of the system, in general, changes after the measurements. The wave function of the system, in general, changes after a measurement.

• So quantum mechanics does not really describe the system, but the information that the rest of the world can possibly have about the system.

Measurement: “Collapsing the Wave Function” into an eigenstate

• In quantum mechanics, a measurement of an observable yields a value, called an eigenvalue of the observable. Many observables have quantized eigenvalues, i.e. the measurement can only yield one of a discrete set of values. Right after the measurement, the state of the system is an eigenstate of the observable, which means that the value of the observable is exactly known.

• A state can be a simultaneous eigenstate of several observable, which means that the observer can exactly know the value of several properties of the system at the same time and make exact predictions about the outcome of measurements of those properties. But there are also incompatible observables whose exact values cannot be known to the observer at the same time. A state cannot be a simultaneous eigenstate of incompatible observables. If it is in an eigenstate of one of the incompatible observables and the value of this observable is known, then quantum mechanics gives only the probabilities for measuring each of the different eigenvalues of the other incompatible observables. The eigenstate of the first observable is a superposition of eigenstates of any of the other incompatible observables. The outcome of a measurement any of the other incompatible observables is uncertain. A measurement of one of the other incompatible observables changes the state of the system to one of its eigenstates and destroys the information about the value of the first observable.