Everyone Pick Up: Syllabus Two homework passes 8/26 Eric Carlson “Eric” “Professor Carlson” Olin 306 Office Hours always (o) (c) Materials Quantum Mechanics, by Eric Carlson (free on the web) Calculator Pencils or pens, paper Physics 741 – Graduate Quantum Mechanics I

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Reading Assignments / The Text ASSIGNMENTS DayReadHomework Today1A, 1B none Friday1C, 1D, 1E1.1 Monday2A, 2B, 2C1.2, 1.3, 1.4 I use my own textbook Downloadable for free from the web Assignments posted on the web Will be updated as needed Readings every day

Homework ASSIGNMENTS DayReadHomework Today1A, 1B none Friday1C, 1D, 1E1.1 Monday2A, 2B, 2C1.2, 1.3, 1.4 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

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Grades, Pandemic Plans Percentage Breakdown: Homework50% Midterm20% Final30% Grade Assigned 94% A 77% C+ 90% A-73% C 87% B+70% C- 83% B<70% F 80% B- Some curving possible Pandemic Plans If there is a catastrophic closing of the university, we will attempt to continue the class: Emergency contacts: Web page Cell:

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 1. Introduction 1A. Quantum Mechanics is Weird Light moves at the speed of light c: A formula from relativity for massless particles: Put these formulas together: Mathematically simpler (and crucial for quantum) to combine them into complex waves: Photons and Quantum Mechanics

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 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 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 Let’s send photons through a half-mirror Detectors A B 50%

Interferometry 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 Now use two mirrors and two half-mirrors We can reconstruct the original waves A B 100% 0% The “phase difference” lets it remember which way it was going Always in detector B 0% 100% Interferometry requires that we carefully position the mirrors

Non-Interferometry How does the photon remember which way it was going? Replace one mirror with a detector A B 25% The “memory” of which way it was going is in both halves C 50% 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 Depending on which experiment you do, photons sometimes act like particles and sometimes act like waves

Can We Have Our Cake and Eat it Too? 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 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 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? A B 100% 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 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 Deriving the free Schrödinger Equation in 1D

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 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 Probability Density

The probability density must be integrated to find the probability that a particle is in a certain range Using Probability Density, and Normalization 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);

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)  e i   (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 Some comments

1D. Fourier Transform of the Wave Function 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 The Fourier Transform In 1D *The factors of 2  differ based on convention. I like this one. Suppose we have a plane wave: It’s momentum is  k 0 It’s Fourier transform is You would expect, if the Fourier transform is concentrated near k 0, then its momentum would be around  k 0 This suggests the Fourier transform tells you something about what the momentum is

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 The Fourier Transform and Momentum Neither of these two representation is really more fundamental than the other We can similarly do Fourier transforms in 3D

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 Expectation Values For example, we can measure the average value of the position x or the position squared x 2 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)

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

We can often gain a qualitative understanding of what is going on by considering the uncertainty principle What stabilizes the Hydrogen atom? Understanding from Uncertainty 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  p from the 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