3 February 2010Modern Physics II Lecture 41 University of San Francisco Modern Physics for Frommies II The Universe of Schrödinger’s Cat Lecture 4

3 February 2010 Modern Physics II Lecture 4 1

University of San Francisco

Modern Physics for Frommies II

The Universe of Schrödinger’s Cat

Lecture 4


Agenda• Administrative matters• Quantum Mechanics

– The Wave Function and its Interpretation– Heisenberg Uncertainty Principle

• Probability vs. Determinism– The Schrödinger Equation in 1-D

• Time Independent Form• Time Dependent Form

– Free Particles; Plane Waves and Wave Packets– Particles and Potentials

• Infinite Square Well (Rigid Box)• Finite Well• Barrier Tunneling• Harmonic Oscillator


First Physics and Astronomy Colloquium:

Wednesday, 10 February 2010 at 4 PMProfessor Richard Muller, Department of Physics, UC BerkeleyRefreshments at 3:30 PMHarney Science Center Room 127

Administrative Matters


TITLE: Physics for Future Presidents SPEAKER: Professor Richard Muller (Dept. of Physics, UC Berkeley and Lawrence Berkeley Lab) ABSTRACT:

Yes, the title of the colloquium is serious. Energy, global warming, terrorism and counter-terrorism, nukes, internet, satellites, remote sensing, ICBMs and ABMs, DVDs and HDTVs -- economic and political issues increasingly have a strong high tech content. Misjudge the science, make a wrong decision. Yet many of our leaders never studied physics, and do not understand science and technology. Physics is the liberal arts of high tech. Is science too hard for world leaders to learn? No. Think of an analogous example: Charlemagne was only half literate. He could read but not write. Writing was a skill considered too tough even for world leaders, just as physics is today. And yet now most of the world is literate. Many children learn to read before kindergarten. We can, and must achieve the same level with scientific literacy, especially for our leaders. Can physics be taught without math? Of course. Math is a tool for computation, but it is not the essence of physics. We often cajole our advanced students, "Think physics, not math!" You can understand and even compose music without studying music theory, and you can understand light without knowing Maxwell's equations.

For the last ten years I have been teaching Physics for Future Presidents at Cal. In that time it has grown from an enrollment of 34 to over 500, and twice has been voted "Best Class at Berkeley." The goal of my course is not to create mini-physicists, but to give future world leaders the knowledge and understanding that they need to make decisions. If they need a computation, they can always hire a physicist. But the knowledge of physics will help them judge, on their own, if the physicist is right. In this lecture I will give 45 minutes of physics that world leaders need to know, and can understand.


The Atom of Bohr KneelsSuccesses:

Predicts the correct Rydberg constant for alkali atoms and things like He+ and Li++. Replace e with Zeffe in the Coulomb force

Failures:

Is successful only for single electron atoms

Fails to explain the “fine structure” of spectral lines, even for alkali atoms

The ground state of Hydrogen has L=0 not 1

Cannot explain the bonding of atoms into molecules.


The Bohr model is an ad hoc theory which fits the hydrogen spectrum. It is a semiclassical theory.

We now know that it does not correctly describe atoms. This description requires a true quantum mechanical theory.

Important 1st step from a purely classical theory to a quantum mechanical one.

“appeared to me like a miracle and appears as a miracle even today.” - A. Einstein, ca. 1940


Quantum Mechanics

Erwin Schrödinger

1887 - 1961

Werner Heisenberg

1901 - 1967

de Broglie proposes matter waves in 1923

Less than two years later two comprehensive theories were independently developed by Schrödinger and Heisenberg.


Two Tracks

Discrete energy exchange

Blackbody Radiation

Bohr Model

Heisenberg’s Matrix Mechanics

Wave Particle Duality

Photoelectric Effect

Matter Waves

Schrödinger’s Wave Equation Max Born’s Probabilty Waves

Paul Dirac showed that the two approaches are entirely equivalent

Nobel prizes: 1932 to Werner Heisenberg

1933 shared by Erwin Schrödinger and Paul Dirac


Paul Adrian Maurice Dirac 1902-1984

Max Born 1882-1970

Max Born was awarded the Nobel in Physics in 1954

Better late than never


The Wave Function and its Interpretation

Important properties: Wavelength, frequency and amplitude

If more than one wave, relative phases become important.

For electromagnetic waves:

Wavelength (or frequency) is a measure of the energy of the corresponding photon.

“Displacement” is the strength of the electric (or magnetic) field. The intensity (btightness) is proportional to the square of these displacements.

For the particle picture, the intensity is just the photon density so

c f

2 2 n E n B


Now recall the discussion of the double slit when we made the light beam very weak so that we were counting one photon at a time.

Over time, as we accumulate counts a distribution of detected photons builds up which is identical to the intensity distribution obtained from the wave picture. Thus, E2 is a measure of the probability of finding a photon at that location.

Now consider matter waves where h

p

The displacement is described by a wave function, x,t), as a function of time and position.

2

2

-

, Probability of finding particle in volume s

Normalization Condi

urrounding ,

tion: ,

.

1x t dV

x t dV dV x t


If we treat particles (including photons) as waves then or E or B) represents the wave amplitude.

If we treat them as particles we must do so on a probabalistic basis

2, Probability / volume of finding particle at ( , )x t x t

We cannot predict, or even follow, the path of a single particle through space and time


Heisenberg Uncertainty Principle

Expectation: By using more precise instrumentation, the uncertainty in a measurement can be made indefinitely small.

Quantum Mechanics: There is a limit to the accuracy of certain measurements. Not an instrumental restriction but an inherent fact of nature.-Ray Microscope and some estimates:

Try to measure the position of an e-

Requires the scattering of at least one , transferring some momentum to the e-.

Greater precision (smaller x) requires shorter Shorter => higher momentum and hence the higher the possible momentum transfer to the electron (higher p).

Feeling pool ball in the dark


The act of observing produces an uncertainty in either or both the position or momentum of the electron. Heisenberg 1927

Let’s make an estimate of the magnitude of this effect. x

Suppose the e- can be detected by a single having momentum h

p

Some or all of this momentum will be transferred, but we can’t tell beforehand how much. Therefore the final e- momentum is uncertain by

hp

x p h Multiplying we obtain


More careful calculations, e.g. using Fourier analysis show that the very best we can do is

2x

hx p

Another form of the uncertainty principle relates energy and time

2

hE t

x<x>

X

x x


Philosophic Implications: Probability vs. Determinism

Newtonian view: xiI

viI

Fi(t)

xiF

viF

Paths can be completely determined

Quantum view: Particles all prepared in the same way will not all end up in the same place.

Q.M. => Certain probabilities exist tat a particle will arrive at different points.

The position and velocity of an object cannot even be known accurately at the same time.


Niels Bohr argued that a space-time description of atoms and electrons is noe possible. Yet, a description of experiments on atoms and electrons must be given in these terms and other familiar concepts, e.g. waves and particles. We must not let our descriptions in such terms lead us to => that these entities actually move in space time as particles.

The Copenhagen interpretation

Correspondence Principle:

Where quantum theory overlaps with the macroscopic world it should predict classical results.

Bohr atom: n=1 and n=2 orbits - Sizes and energies quite different

n=105 and n=105+1 – Very close in size and energy. Tends to a continuum of orbits which is the classical picture.


2

12n

ZE E

n


The Schrödinger Equation in One Dimension

Matter wave function Like any wave function it should satisfy a wave equation. The equation for electromagnetic waves in 1-D is 2 2

2 2 2

10

t

E E

x c

Erwin Schrödinger was challenged to come up with such an equation for matter waves and did so during a secluded weekend with a “friend”.

The Schrödinger equation cannot be derived but had to be invented, just as Newton invented F = ma.

The Schrödinger equation can be written in two forms, time- dependent and time-independent. For stationary states the time-independent solutions, , will suffice.


The Schrödinger equation can be justified as a statement of conservation of energy. In one dimension we have

2 2

2

( )( ) ( )

2

xU x x E x

m x

Again, this equation is not derived. It does however aexpress energy conservation and its solutions give results in accord with experiments for a wide range of situations.

Requirements for Solutions:

Continuous - no sudden jumps

Normalized – probability of finding particle somewhere in all space is exactly 1 (or 100%)


Free Particles; Plane waves and Wave Packets

Particle that is not subject to any force, so we can set U = 0

Simplest solution: plane wave (single

( ) sin or cosx A kx B kx

Note that this wave extends indefinitely, from - to . This is consistent with H.U.P. since a single => we know p = h/exactly => the position, x, must be totally unpredictable.

Problem with normalization: The sum of |(x)|2 = A2sin2(kx) over all x is infinite for all nonzero values of A.

What to do? Our “lovely” (?) probabilistic interpretation of 2 has self-destructed for the simplest case of a plane wave.


There are 2 ways out of this dilemma:

(1) Give up the idea of absolute probability when dealing with wave functions that are not square integrable.

(x)2 dx is then interpreted as the relative probability of

finding the particle in dx at x2

12

2

( )

( )

x dx

x dx

Relative probability of finding particle near r1 to that of finding it near r2

(2) Give up the idea of a free particle with perfectly defined momentum and superpose plane waves of different momenta to form a localized wave packet which can be normalized.


x

Note that x now has some finite spread as does p

xx p

The sum of |y(x)|2 = A2sin2(kx) is now finite so that a nonzero value of A can be chosen to make this sum exactly 1.


Is this superposition of waves of different wavelengths (or frequencies) legal?

Yes, the Schrödinger equation is what is called a linear differential equation. For such an equation, any linear combination of solutions is also a solution.

1 1 2 2 3 3a a a



Particles and Potentials

Infinite Square Well (Rigid Box):

sin cosx A kx B kx

In the region 0 < x < L the particle is free

Outside the well U = 0 so

must be continuous so L) = 0

0 and sinB A kx Boundary conditions imposedAlso

0 0,1,2,L kL n n

k and hence E can have only certain values

22

28

nk

L

hE n

mL


2

1 28

hE

mL is the ground state of a particle in a box

E1 ≠ 0 zero point energy

Still there even at 0ºK

Wave functions

2sinn

nx

L L

2 2

01 * sin

L ndx A x dx

L

No plane wave troubles because = 0 outside well.



Uncertainty in the Box

221

1 2 Ground state energy:

8 2

phE

mL m

1momentum: 2

hp

L

1

We don't know the direction of motion

p=2h

pL

Particle can be anywhere in box x L

hp x L h

L

Consistent with H. U. P.


U

0 Lx

U0

0

( ) 0 0

( ) 0 and

U x x L

U x U x x L

0

II

Inside the well (II), if

we again have a free particle

sin cos

E U

x A kx B kx

I II III

At 0 and we can no longer just set 0L

Finite Square Well

In I and III the wave function decreases exponentially with distance from the well.

We do however still insist upon ’s continuity across the well boundaries.


Inside well: Sinusoid Ground state E < well

Outside well: exponential decay of

0 outside is classically forbidden.


We appear to have violated conservation of energy

Look to the uncertainty principle in the form

2

hE t

This tells us that E is uncertain and can even be nonconserved for very short times t h / E

Particle with E > U0 is everywhere sinusoidal

changes as the particle transits the well.

For E > U0 , any E is possible

For E < U0, E is quantized


Tunneling

0

2

2

0

Classically, Regions I and III: 2

Region II: 2

E U

mvE

mvE U

0

0

Classical mechanics allows no reflection for

and requires total reflection for

E U

E U


Quantum mechanics adds the word “almost” to this picture.

0

0

0

: almost total transmission

: almost total reflection

When E is comparable to U unusual nonclassical

phenomena may occur.

E U

E U

Remember that in quantum mechanics we are dealing with probabilities



2

0

2

2where

GLT e

m U EG

Increasing the barrier height or width rapidly decreases T. Correspondence principle returns us to classical picture.


Example: 50 eV electron approaches a square barrier 70 eV high and (a) 1.0 nm thick, (b) 0.10 nm thick. What are the transmission probabilities?

19 180(a) 70 eV 50 eV 1.6 10 J/eV 3.2 10 JU E

32 18

92-34

2 9.11 10 kg 3.2 10 J2 2 1.0 10 m 46

1.06 10 J

s GL

2 2 201. 1 1 0GL GLT e e

4.6

(b) For 0.10 nm, 2 4.6

0.010 or 1%T e

L GL


A model for decay is 2p and 2n so q = 2e 4He nucleus


3-D Infinite Well (crystal lattice)

Lx

Ly

Lz

2

2

2 2 22

2 2 2

( , , ) , ,2

where

x y z E x y zm

x y z

Separation of variables:

Assume a solution ofthe form , ,x y z X x Y y Z z


22 22 2

2 2 22yx z

x y z

nn nE

m L L L

Note that we now have 3 quantum numbers, one for each degree of freedom.

2 2 2 2

2 2 21 2 2

1st excited state: , , = 2,1,1 or 1, 2,1 or 1,1,2

3 2 1 1

2

x y z

st

n n n

EmL mL

More than one solution for a given E state is degenerate

Cube, all lengths the same


Simple Harmonic Oscillator

Some systems may be approximated by a quantum mechanical 1-D oscillator, e.g. a vibrating diatomic molecule.

Any system in a potential minimum behaves approximately like a SHO.

2

Hooke's law force:

1and its associated potential energy:

2 spr

ing consta

nt

F kx

U kx

k



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3 February 2010Modern Physics II Lecture 41 University of San Francisco Modern Physics for Frommies II The Universe of Schrödinger’s Cat Lecture 4