Physics 741 – Graduate Quantum Mechanics I

Everyone Pick Up: • Syllabus • Two homework passes Materials • Quantum Mechanics, by Eric Carlson (free on the web) • Calculator • Pencils or pens, paper Physics 741 – Graduate Quantum Mechanics I Eric Carlson “Eric” “Professor Carlson” Olin 306 Office Hours always 758-4994 (o) 407-6528 (c) ecarlson@wfu.edu http://users.wfu.edu/ecarlson/quantum/index1.html 8/27

Dr. Carlson’s Approximate Schedule Monday Tuesday Wednesday Thursday Friday 9:00 10:00 research research research 11:00 12:00 PHY 741 office hour PHY 741 PHY 741 office hour 1:00 office hour PHY 215 PHY 215 office hour PHY 215 2:00 research research research 3:00 Free food 4:00 Collins Hall colloquium 5:00 • I will try to be in my office Tues. Thurs. 12-2 • Also in Collins Hall Tues. 3:30-5:00 • When in doubt, call/email first

Reading Assignments / The Text http://users.wfu.edu/ecarlson/quantum/index1.html • I use my own textbook • Downloadable for free from the web • Assignments posted on the web • Will be updated as needed • Readings every day ASSIGNMENTS DayReadHomework Today 1A, 1B none Wednesday 1C, 1D, 1E 1.1 Friday 2A, 2B, 2C 1.2, 1.3, 1.4

Homework http://users.wfu.edu/ecarlson/quantum • Almost all the problems in the textbook • About two problems due every day • Homework is due at 12:00 on day • Late homework penalty 20% per day • Two homework passes per semester • Working with other student is allowed • Seek my help when stuck • You should understand anything you turn in ASSIGNMENTS DayReadHomework Today 1A, 1B none Wednesday 1C, 1D, 1E 1.1 Friday 2A, 2B, 2C 1.2, 1.3, 1.4

Attendance and Tests Tests • Midterm will be about two hours long, approximately on Oct. 19 • Final exam will be about three hours long, 9:00 AM on Dec. 12 Attendance • I do not grade on attendance • Attendance is expected • Class participation is expected • I take attendance every day

Grades, Pandemic Plans Grade Assigned 94% A 77% C+ 90% A- 73% C 87% B+ 70% C- 83% B <70% F 80% B- Percentage Breakdown: Homework 50% Midterm 20% Final 30% • Some curving possible Emergency contacts: Web page email Cell: 336-407-6528 Pandemic Plans • If there is a catastrophic closing of the university, we will attempt to continue the class:

1. Introduction 1A. Quantum Mechanics is Weird Photons and Quantum Mechanics • Light come in packets with energy • This equation is bizarre • It implies both that EM energy is in waves (f) and is particles (E) • A plane wave looks something like • Mathematically simpler (and crucial for quantum) to combine them into complex waves: • Light moves at the speed of light c: • A formula from relativity for massless particles: • Put these formulas together:

Uncertainty Principle for Photons • Waves are not generally localized in space • They have a spread in position x • You can make them somewhat localized by combining different wave numbers k • Now they have a spread in k, k • There is a precise inequality relating these two quantities • Proved formally in a later chapter • If we multiply this equation by , we get the uncertainty principle • In quantum mechanics, photons can’t have both a definite position and a definite momentum

It’s All Done with Mirrors • Ordinary mirrors reflect all of the light that impinges on them • Half-silvered mirrors have a very thin coating of metal • They reflect only half the light and transmit the other half • What happens if we shine photons on a half silvered mirror? mirror half-mirror

Photons and half-silvered mirror Let’s send photons through a half-mirror • The photon gets split into two equal pieces • Each detector sees 50% of the original photons • Even if we send photons in one at a time • Never in both detectors 50% A 50% • If you send in a wave the other way, the same thing happens • There’s a “phase difference”, but since we square the amplitude, the probabilities are the same • 50% in each detector Detectors B 50% 50%

Interferometry Now use two mirrors and two half-mirrors • We can reconstruct the original waves • The photon gets split into two equal pieces • The two halves of the photons are recombined by the second half-mirror • Always goes to detector A • Even one photon at a time • If you send in a wave the other way, the photon is still split in half • The “phase difference” lets it remember which way it was going • Always in detector B 0% A 100% Interferometry requires that we carefully position the mirrors 100% B 0%

Non-Interferometry How does the photon remember which way it was going? • Replace one mirror with a detector • The photon gets split into two equal pieces • Half of them go to detector C • The other half gets split in half again • Detectors A and B each see 25% • Even if you do it one photon at a time • The “memory” of which way it was going is in both halves A 25% C • Depending on which experiment you do, photons sometimes act like particles and sometimes act like waves 50% B 25%

Can We Have Our Cake and Eat it Too? • When you do interference, you can tell the photon went both ways • For other experiments, you can measure which way it went • Can we do both? The plan: • Do experiment in space (no friction, etc.) • Carefully measure momentum of mirror before you send one photon in • Check photon goes to detector A • Remeasure momentum and determine the path A 100% The problem • If you measure the mirror’s initial momentum accurately, you have small p, and big x • Poor positioning of mirror ruins the interference B 0%

Conclusion • Because Quantum applies to photons, it must apply to mirrors as well • It seems inevitable that it applies to other things (like electrons) • We will apply it to non-relativistic systems, because these are easier to understand than relativistic • We will eventually do electromagnetism / photons, but this is harder

1B. Schrödinger’s Equation Deriving the free Schrödinger Equation in 1D • For electromagnetic field, we had waves like: • For non-relativistic particles, we rename the wave function: • We assume two other relationships still apply • We note that: • Multiplying by , we have: • Classical relationship between the energy E and momentum p: • Multiply by  on the right: • Make the substitutions above

Schrödinger’s Equation in 1D • What if we have forces? • If the forces conserve energy, they can be written as the derivative of a potential • This contributes a new term to the energy • Make the same substitutions as before

Schrödinger’s Equation in 3D • What if we are in 3D? • Momentum and potential must be generalized: • Redo the calculation as before • Make the new substitutions:

Other Things to Consider: • Particles may have spin • Some forces (such as magnetism) are not conservative forces • There may be multiple particles • The number of particles may actually be indefinite • Relativity • We will deal with all of these later

1C. The Meaning of the Wave Function Probability Density • The wave function, in essence, describes where the particle is • When it’s zero, the particle isn’t there • When it’s large, the particle more likely is there • In electromagnetism, the energy density is proportional to |E(r,t)|2 • Since the probability of finding a photon is proportional to the energy density, it makes sense to make a similar conclusion for wave functions

Using Probability Density, and Normalization • The probability density must be integrated to find the probability that a particle is in a certain range What are the units of (x,t)? Of (r,t)? • The probability that a particle is somewhere must be exactly 1 • Don’t forget how to do 3D integrals in spherical coordinates!

Sample Problem A particle in a three-dimensional infinite square well has ground state wave function as given by What is the normalization constant N? What is the probability that the particle is at r < ½R? > integrate(sin(Pi*r/R)^2,r=0..R);> integrate(sin(Pi*r/R)^2,r=0..R/2);

Some comments • Schrödinger’s equation is first order in time • (r,t= 0) determines (r,t) at all times • Phase change in (r,t) does not affect anything • (r,t)  ei (r,t) is effectively equivalent • We will treat these as different in principle but experimentally indistinguishable • Normalization condition must be satisfied at all times • Can be shown that if true at t = 0, Schrödinger’s equation makes sure it is always true • Will be proven later • We still need to discuss how measurement changes (r,t) • Chapter 4

1D. Fourier Transform of the Wave Function The Fourier Transform In 1D • The Fourier transform of a function is given by* • You can also Fourier transform in reverse • If the function is normalized, so is its Fourier transform • Suppose we have a plane wave: • It’s momentum is k0 • It’s Fourier transform is • You would expect, if the Fourier transform is concentrated near k0, then its momentum would be around k0 • This suggests the Fourier transform tells yousomething about what the momentum is *The factors of 2 differ based on convention. I like this one.

The Fourier Transform and Momentum • The wave function (x) tells you the probability of finding the particle’s position x in a certain range • The Fourier Transform tells you the probability of finding the particle’s momentum p in a certain range • Neither of these two representation is really more fundamental than the other • We can similarly do Fourier transforms in 3D

Expectation Values • In any situation where you have a list of probabilities of a particular outcome, the expectation value is the average of what you expect to get • If there is a continuous distribution of possibilities, this becomes an integral • For example, we can measure the average value of the position x or the position squared x2 • Using the Fourier transform, we can similarly compute the momentum or its square • There are in fact ways of finding these without doing the Fourier transform (later chapter)

Uncertainty • The uncertainty of a quantity is the root mean square average of how far it deviates from its average • There is an easier way to calculate this: • Uncertainty in position are given by: • These satisfy:

Understanding from Uncertainty • We can often gain a qualitative understanding of what is going on by considering the uncertainty principle • What stabilizes the Hydrogen atom? • It has potential and kinetic energy: • Classically, it wants: • p = 0 to minimize the kinetic energy • r = 0 to minimize the potential energy • But this violates the uncertainty principle! • If we specify the position too well, the momentum will be large • If we specify the momentum too well, the position will become uncertain • A compromise is the best solution

Sample Problem Classically, the Hydrogen atom has energy given by Using the uncertainty principle, estimate the ground state energy of the hydrogen atom • Let the uncertainty in the position be • By the uncertainty principle, the momen-tum must have at least uncertainty • Assume the position (r) is about x from the ideal position (r = 0), and the momentum (p) is about pfromthe ideal momentum (p = 0) • Substitute into the energy formula • The minimum is at some finite value of a: • Substitute back into energy formula • Off by factor of 4 • Mostly because we ignored that it was a 3D problem (factor of 3) • Partly because it’s an estimate

Physics 741 – Graduate Quantum Mechanics I