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Transcript

This course is: • Fun! – The ideas are presented in a way that makes a LOT more sense than a usual QM course – Focus on understanding one exemplar experiment in detail – you will understand it well – The simulation is pretty neat and allows you to relate the math to ‘real’ measurements • Hard! – New notation and language – New math (hopefully not bad thanks to a great preface) – Ideas that take some getting used to! • Suggestions: – – – – – Make a vocabulary list – each bolded term in the text is a good place to start Make a list of symbols and what they mean Write the postulates in your own words with examples and explanations Draw diagrams for any situation which is at first unclear to you Do/repeat calculations to get practice Goals: • Understand the 6 postulates of quantum mechanics (QM) and the nature of QM • Proficiently use matrix mechanics in the context of QM • You will be able to solve 4 fundamental types of problems – each done in detail within a specific context, then generalized to any system: – Sequential Stern-Gerlach measurements • Analyze generic QM problems using matrix mechanics – Use time evolution to understand spin precession • Analyze generic time dependent QM problems using matrix mechanics Postulates of QM •What is a postulate? •To assume or claim as true, existent, or necessary (from MerriamWebster) •Tell us how to treat a QM system •Successful and tested ideas •Can not be proven •Accepted as accurate but would need to be re-evaluated if new results contradicted them •6 main postulates •Right now they have limited meaning •Given so you know where we are headed and connect to preface math 6 main postulates (and examples) 1. 2. 3. 4. 5. 6. states are defined by kets: |+>, |-> Operators are matrices: H, S… We measure eigenvalues: +/- h/2 Probability is found by: |<+|psi>|2 States “reduce” (collapse) |psi> goes to |+> Schrodinger equation gives time evolution, for instance, Larmor precession 1.2 Stern-Gerlach experiment Neutral silver atoms pass through a region of inhomogeneous magnetic field •Why silver? Why neutral? •Why an inhomogeneous field? Found two locations where the silver atoms ended up •Why is this odd? S Ag oven Non uniform B-field N magnet screen Ag: 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 4d10 5s1 Number of electrons 46 Angular momentum = ? + 1 S = 47 S-G historic overview • They thought there should be splitting with the Bohr model because they thought that the silver atom should have a h/2pi orbital angular momentum from that model, when in fact it's zero - L not zero, should see splitting (by S-G), and after much effort, they do and conclude Bohr is right - but why splitting and not uniform if initial orientations are random? (Einstein and Ehrenfest) - QM evolves, Bohr model found inadequate - wait, L IS zero, why did they see splitting? - then 5 years later, the idea of intrinsic spin... Consider our expectations on what should happen to a neutral particle in an inhomogeneous magnetic field: •What does a magnetic field interact with? •How can a neutral atom interact with a magnetic field? •Let’s derive it classically from intro-course principles •What does a simple magnetic dipole look like? •What does the energy look like? •What will the force be and why does the B need to be inhomogeneous? •How do we relate this to angular momentum? •Why do we introduce “spin”? •Does it really “spin”? •What is different between what we expect to observe classically and what we actually observe? •What is a projection? •What does two “spots” tell us about the spin? •What is quantization? “that” calculation • More rigorous details posted on blackboard • Found μ = (q/2m)L • Have such a term for orbital angular momentum L, “intrinsic spin” S, and for the total angular momentum (the QM sum of L and S) • Generalize: μ = g(q/2m)S – S is the “intrinsic angular momentum” – as if the electron spun on its axis, but NOT physical – g is the gyromagntic/gyroscopic/g-ratio – g is dimensionless – g for electron is one of best known values in physics What is “intrinsic spin”? • Also called “spin”, or spin angular momentum, or S • It’s a “degree of freedom”, or quantum number: a “state” the particle has • Does interact with magnetic fields like L, but not continuous! • NOT a physical rotation • INTRINSIC property – like charge and rest mass! We have no model for what “makes it up/causes it” for fundamental particles • Shows up most simply in Pauli exclusion principle Stern-Gerlach Experiment 1 •What do we find here? •First SG device is a “state preparation device” or “polarizer” •2nd device is the “analyzer” •How can we write the state vector out of the 1st device? •Out of the 2nd device? x Z 0 Z -x - 0 0 Stern-Gerlach Experiment 2 What do we see? If we do this one atom at a time, can we predict where it will end up? What can we conclude about the relationship between the plus z state and the x states? Write a possible vector to show that relationship LETS do the math for this – probabilities! x Z 0 X -x - 0 0 Stern-Gerlach Experiment 3 What do you expect to see? TRY IT! (a) Z X x Z - 6 main postulates (and examples) 1. 2. 3. 4. 5. 6. states are defined by kets: |+>, |-> Operators are matrices: H, S… We measure eigenvalues: +/- h/2 Probability is found by: |<+|psi>|2 States “reduce” (collapse) |psi> goes to |+> Schrodinger equation gives time evolution, for instance, Larmor precession Recap: • Observe deflection of neutral atoms in inhomogeneous magnetic field AS IF they had angular momentum L: – Except L is zero for those atoms – must be some other “source” of angular momentum – EXCEPT that instead of a continuous distribution of deflection we get ONLY 2 “SPOTS” (“Quantization!”) – Propose “S” = spin = intrinsic “angular momentum” – Measuring the “strength” of that interaction (‘g’) is of fundamental importance to particle physics research – L and S can be combined quantum mechanically and will be of importance to you later in your QM studies Recap: • S-G apparatus is how we look at the deflections – the simulation is an easy way to make “measurements” with successive S-G apparatuses at different orientations – Observe Z-Z measurement “prepares” the state with spin along Z either “up” or “down”, and then “measures” the state to still be as it was prepared – Observe Z up and Z down (or X or Y) are orthonormal – Observe Z-X (and Z-Y, and X-Y…) measurements and find a probability that half the atoms with “spin up in Z” (or down) will have “spin up in X” (or down) – Postulate |+> = a|+>x + b|->x where a and b are complex, a*a = b*b = ½, and a*a + b*b = 1 Conventions for “states” • Can’t exactly specify the states, but have “conventional” choices (details in text): • (WRITE THESE ON THE BOARD!!) • Row 1: write |+> and |-> in terms of |+>y and |->y • Row 2: show that the |+>y and |->y states are orthonormal • Row 3: write |+>x in terms of |+>y and |->y “Unexpected” observation: • Saw that Z-X-Z did not “preserve” the spin up Z states as spin up • SAME thing happens with polarization of light! This is not “bizarre” but is a property of a 2-state system of this type – Maybe: The S-G devise to analyze the spin states along “X” must “alter” the state somehow (analogy with polarizer – it “rotates” the polarization – or more accurately the projection is non-zero – Maybe: There is some effect from Heisenberg Uncertainty Z X Z - Calculations: • ROW 1: Using our new knowledge about how to write the spin states, calculate the probability that a state initially with spin up along Z will have spin up along Z after passing through a S-G devise for analyzing the spin-X state (the “Z-X-Z” system) • ROW 2: do the same for spin up along Z ending up with spin down along Z • ROW 3: do the same as ROW 1 except for the “ZY-Z” system • (if you finish early work on another one!) Stern-Gerlach Experiment 4 Consider what you found with the Z-X-Z system (shown in the top figure) What do you expect to see if we combine the output beams from both ports on the X devise, and put them into a Z? TRY IT! CALCULATE IT! What state are we putting into the last S-G device? Z X Z Z - x X -x Z - Stern-Gerlach Experiment 4 - continued • “you open the window further and let more sunlight in yet the room gets darker!” • Book discusses this by analogy with young’s double slit interference experiment – what is the similarity? z electron beam I1 F1 z I I I1 I 2 F2 I2 • We can also explain this effect using vector addition, and by analogy with polarization vectors: + = combination of states (“mixture”) vs. superposition • |+>x = (1/√2)[ |+> + |-> ] • IF this meant half the atoms were in |+> and half were in |->, what would we get if we put this state into another S-G devise to measure the spin along X? • How do we know which we have in our S-G system? • (CALCULATE IT!) Matrix notation: DO: x<-|->y in matrix notation for practice! Dirac: • Who was P.A.M. Dirac? – Paul Adrien Maurice Dirac (19021984) – Studied general relativity and the then brand new quantum mechanics – Quote “In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in poetry, it's the exact opposite” – How did Dirac get his wife? 1.5 General Quantum Systems – more than 2 outcomes Kronecker Delta: dij = {1, i = j 0, if i ≠ j } so Σi dijai = aj Summation: Σi bi|ai = b1|a1 + b2|a2 + b3|a3 + … Where the b’s are complex, by convention b1 is real Given: observable A – THIS IS A HERMITIAN MATRIX! eigenvalues λn eigenstates an orthonormality ai aj = dij completeness y = Si aiy ai = Si aiaiy Si aiai = 1 Conjugation: yf = fy* General Quantum Systems • Other 2-level systems: – All fundamental fermions (leptons and quarks) are spin-1/2 systems – Qubit (0 and 1 bit for quantum computing) candidates: 2 sets of polarization pairs for photons, spin up and down of electron, electron’s position in a quantum dot, atoms or ions either in spin states, or two hyperfine (from intrinsic spin inside the nucleus) levels • Other small-N-level systems: – All baryons (like the proton) are composed of 3 spin-1/2 quarks, and this can be spin 1/2, 3/2, … – Bosons (force carriers) have integer spin: photon is spin 1, graviton (if it exists) is spin 2 • Large/infinite-N systems: (usually represented by a wave function) – Finite quantum well – Harmonic oscillator Stern-Gerlach Experiment 4 What do you find as the output if you run all atoms through at once? What do you find if you observe one atom at a time?? LOOK AT VIDEO FOR PHOTON ANALOG! “Interference” is a property of states being SUPERPOSITIONS – NOT mixtures What do you think it means to “COLLAPSE” a state?? (postulate 5) Z x X -x Z - Statistical Quantity Reminders 1 x= N Mean: data N x i =1 i Number of data Standard deviation: s = 1 N 2 x x = i N - 1 i =1 Standard deviation of the mean: Probability: P= x M 1 N 2 N 2 x x i N - 1 i =1 N -1 s m = N Number of atoms Standard deviation of the probability: p = m M