It’s fun!

1. 11. Interpretation of Quantum Mechanics Why Discuss Philosophy? • It’s fun! • Different ways of thinking about quantum mechanics sometimes involve different calculation techniques • Sometimes, these techniques make problems easier My goals: • Teach you useful techniques • Show you how to use them to solve difficult problems • Explode your brains

2. 11A. The Time Evolution Operator Definition • The state vector |(t) is a linear function of the initial state vector |(t0) • Call the operator that performs this function: • This operator must preserve probability, so it must be unitary • Some other easy-to-prove identities: Schrödinger’s Equation for U: • We know that • Therefore • This, together with the boundary condition U(t0,t0) = 1 defines U(t,t0)

3. Linearity of Time Evolution operator • Suppose we have two solutions of Schrödinger’s Equation, |1 and |2 • We know Schrödinger’s equation is linear, so thatis also a solution • We thereforeknow that • We see that U is a linear operator • It is also reversible • Measurement, in contrast, is neither linear nor reversible • Clearly, the time evolution operator only applies when not performing measurements

4. Finding the Time Evolution Operator • If H is independent of time, easy to solve these equations: • Suppose we have a complete set of orthonormal eigenstates of H: • Then insert these states into expression for U: • If H depends on time, expression gets complicated

5. Sample Problem A harmonic oscillator with mass m and angular frequency  in 1D is initially in the state |(t0) at time t0. At a later time t, the energy is measured. What is the probability that it will be measured to have the minimum value ½? • The probability is just • We need to evolve the initial state to the final state: • Now we get clever: let H act to the left:

6. Sample Problem A spin-1/2 particle is in one of the states |+ or |- at time t0. Find a Hamiltonian that evolves it to the state |+ by time t. • We need: • Take the inner product ofthe first with the second • This is impossible, so NO SOLUTION • Does this mean one can never create a spin + particle? • Yes, if this particle is the only particle in the universe, and all it has is spin • If we have some other particle, it is always possible to do something like

7. Sample Problem Consider a superposition detector. This device is initially in the state |S0, but such that when it interacts with a spin |, it will change into state |S-,? when faced with a pure spin state | +  or | –  , and state |S+,? when presented with a superposition state, where “?” means that it may represent any quantum state. Show such a device is impossible • What we want: • We will make no assumptionsabout a, b, c, other than that theS+ and S-states are orthogonal • By linearity: • So impossible

8. 11B. The Propagator It’s Reason for Existence • Consider a single spinless particle (in 3D) • This can be generalized • Given (r,t0), what is (r,t)? • Insert a complete set of states |r0 • Now define the propagator, also called the kernel: • Then: • We can find the propagator, and use it to get the wave function later in one step

9. Schrödinger’s Equation for the Propagator • From this equation, easy to see • Assume we have Hamiltonian • Schrödinger’s Equation • Since true for all(r0,t0), wemust have

10. Propagator for Constant H • By definition, • If H is constant, recall • It follows that • Therefore

11. Propagator for No Potential • Let’s work it out for a free particle in one dimension V(x,t) = 0 • The eigenstates are plane waves • The propagator is then • Each of these ki integrals can be done using:

12. Sample Problem At t = 0, the wave function of a free particle is given by Find (x,t) at arbitrary time • In 1D, the final wave functionwill just be • Now we just substitute in: • Use the identity:

13. Sample Problem At t = 0, the wave function of a free particle is given by Find (x,t) at arbitrary time

14. 11C. The Feynman Path Integral Formalism The Idea Behind it • It is hard to find K for large time differences, but easy for small • We can build up large ones out of many small ones • Consider the Hamiltonian in 1D: • We wish to solve: • For short enough times, we expect V to change relatively little, and K to be non-zero only near x = x0 • Estimate V(x,t) = V(x0,t0)

15. Propagator for Constant H • Multiply by • Cleverly write this as • This is same as equation for free propagator,and has the same boundary condition • It therefore has same solution, at time t1 slightly after t0, at position x1: • Let t = t1 – t0, then we have

16. Wave Function at Time tN • Since U(t2,t0) = U(t2,t1)U(t1,t0), we can get it at t2 = t0 + 2t • Iterate it N times to get it at time tN = t0 + Nt

17. Functional Integrals • In limit t  0, we are considering allpossible functions xi(t) that start at x0and end at xN • Define the functional integral: • The propagator is now:

18. The Lagrangian and the Action • In the limit t  0, the term in round parentheses is a derivative • The inner sum is the value of a function at various times, added up, and multiplied by the time step • An integral • That thing in []’sis the Lagrangian • The integral of theLagrangian is the action

19. Second Postulate Rewritten: • The propagator acts on the wave function to make a new wave function • This can be generalized completely to rewrite the second postulate: • I am being deliberately vague because we won’t ever actually use this version • It is identical with the previous one Postulate 2: When you do not perform a measurement, the state vector evolves according to where S[x(t)] is the classical action associated with the path x(t)

20. Why This Version of the Postulate? • The Lagrangian and action are considered more fundamental then the Hamiltonian • Hamiltonian is normally derived from the Lagrangian • The action is relativistically invariant, the Hamiltonian is not • In quantum field theory, it is far easier to work with the Lagrangian • For some problems in quantum chromodynamics, it is actually the only known way to do the computation Why Not This Version of the Postulate? • To do any problem, you must do infinity integrals – hard even for a computer • I know of no doable problem with this approach

21. Connection with Classical Physics • According to this postulate,to go from xI to xF, the particletakes all possible paths – pretty cool • But which ones contribute the most? • If we consider  small, then almosteverywhere, the phase is constantlychanging for even a slight change of path • Unless small changes in path leave the action stationary • Stationary phase approximation • This is the same as the classical path! xI xF

22. 11D. The Heisenberg Picture Rearranging Where the Work is Done • Quantum mechanics makes predictions about outcomes of measurements • Can be shown: All we need to do is predictexpectation values of operators at arbitrary time • Using the time evolution operator, we relate this to time t0: • In Schrödinger picture, the state vector changes and the operator is constant • Why not try it the other way? • Let the state vector be constant and the operator changes • Define the Heisenberg picture: • Then we have:

23. Evolution of Operators in Heisenberg • Assume an operator in the Schrödinger picture has no time dependance • In the Heisenberg picture, it evolves according to: • Recall Schrödinger’s equation for U: • Hermitian Conjugate of this expression: • Take time derivative of AH:

24. Second Postulate In Heisenberg Postulate 2: All observables A(t) evolve according to where H(t) is another observable. • Note that if A has explicit time dependence, another term must be added • If the Hamiltonian has no explicit time dependence, then H will not evolve, so • Other postulates must be changed slightly as well • State vector does change, but only during measurement

25. Heisenberg vs. Schrödinger • Schrödinger says the state vector is constant, but the operators change • To me, this is counterintuitive, since, for example, it is only in measurement that a particle changes • Since the two have identical predictions, there is no way to know which one is “right” • I think in Schrödinger but will do calculations in whatever is convenient

26. Commutation of Operators in Heisenberg • Suppose we have a commutation relation in Schrödinger: • What is the corresponding commutation in Heisenberg? • Recall: • Abbreviate this: • We therefore have • For example, in 1D, we have • Note that in unequal times, thereis no comparable relationship • At unequal times, many operatorsdon’t commute with themselves

27. Example of Operator Evolution • Consider a free particle in 1D • Let’s find evolution of momentum operator first • And now for position: • Need to solve these two equations simultaneously • Momentum one is easy: • Then we solve position one • Note that • Recall generalized uncertainty principle: • This implies, in this case,

28. Sample Problem How long can you balance a pencil on its tip before it falls over? • Not exactly a real problem, but we’ll tackle it anyway • We need expressions for thekinetic and potential energy • If we treat pencil as a uniformrod of mass m and length L, then • In a manner similar to how this is handled classically, you define the momentum corresponding to , and call it P • Then the Hamiltonian will be • For small angles • So 

29. Sample Problem (2) How long can you balance a pencil on its tip before it falls over? • The angle variable  and itscorresponding momentum satisfy: • Now work out time derivatives of operators:

30. Sample Problem (3) How long can you balance a pencil on its tip before it falls over? • Take second derivativeof  with respect to t: • The solution to this is: • This has boundary values: • Rewrite second equation in terms of P: • Write (t) in terms of (0) and P(0):

31. Sample Problem (4) How long can you balance a pencil on its tip before it falls over? • Look at commutator of  at different times: • Generalized uncertainty relationship: • Therefore:

32. Sample Problem (5) How long can you balance a pencil on its tip before it falls over?  • Typical pencil: • Substitute in: • Order of magnitude: if (0) = 1 or (t) = 1, then the pencil has tipped over • Maximum time for it to balance: • Solve for tmax:

33. The Interaction Picture • Half way between Schrödinger and Heisenberg • Divide the Hamiltonian into two pieces, H0 and H1(t): • Normally, H0 is chosen time-independent and easy to find the eigenstates of • Then operators evolve due to H0 and state vectors due to H1: Why would we do this? • It is a useful way to do time-dependent perturbation theory • We will ultimately use this approach, but not use this notation • It is a useful way to think about things • Very common in particle physics • Think of the state as “unchanging” until the pion decays • We will, in this class, nonetheless always work in Schrödinger picture

34. 11E. The Trace Definition • Let {|i} be a complete orthonormal basis of a vector space  • Let A be any operator in that vector space • Define the trace of A as • In components: • Can be shown: trace is independent of choice of basis: • Consider trace of a product of operators:

35. Partial Trace • A trace reduces an operator in vector space  to a number • If we have an operator A in a product space of vector spaces,  , we can do a trace over just one of them, say , to get an operator on vector space  • Suppose the vector spaces  and  have basis vectors {|i} and {|j} respectively • Basis vectors of   look like {|i,j} • Define the partial trace as • This makes Tr(A) an operator on  • In components, this is

36. 11F. The State Operator / Density Matrix Two Types of Probability • There is a classical sense of probability that has nothing to do with quantum mechanics: • If I pull a card from a deck of cards, the probability of getting a heart is 25% • We don’t believe it is truly indeterminate, just that we are ignorant • Quantum mechanics introduces another kind of probability • If a particle has spin + in the x-direction, and we measure the spin in the z-direction, the probability that it comes out + is 50% • Up to now, we assumed that the quantum state is completely known • What if there are multiple possible quantum states? • Quantum states |i(t) each with probability fi • The probabilities fi arenon-negative and add to one • The quantum states will be normalized, but not generally orthogonal

37. The State Operator • In principle, this list of possible states/probabilities could be very complicated • Define the state operator as Properties of the state operator: • Trace: • Hermitian (obvious) • Positive semi-definite: for any state vector |:

38. Sample Problem An electron is in the spin state + as measured along an axis at an angle randomly chosen in the xy-plane. What is the state operator? • We are in the normalized positive eigenstate of the operator • The normalize eigenstate is: • If we knew whatthe angle  was,the state operator would be • Since we don’t, and all angles are equally likely, we have to average over all angles: • Let’s check we got the trace right:

39. Eigenvectors and Eigenvalues of State Operator • Like any Hermitian operator, we can find a complete,orthonormal set of eigenstates of  with real eigenvalues • Because it is positive semi-definite, we have • Trace condition: •  written in terms of its eigenvectors: • Compare to the definition of : Conclusions: • We can pretend  is a combination of orthonormal states, even if it isn’t • Any positive semi-definite Hermitian operator with trace 1 can form a valid state operator

40. Pure States and Mixed States • If there is only one non-zero i, then we have a pure state • If it isn’t a pure state, it’s a mixed state You can prove something is pure in a variety of ways: • You can find the eigenvalues (homework), and show they are 0 and 1 • You can find 2and compare it to : • For a pure state, i2 = i (because it is zeroor one), for a mixed state it isn’t • One measure of how mixedthe state is is the quantummechanical entropy • Pure states have S = 0, mixed states have S > 0

41. Time Evolution of the State Operator • Whichever state vector it is in, it satisfies Schrödinger’s Equation: • It follows that: • Don’t confuse this with the Heisenberg picture:

42. Expectation Values of Operators • If we knew which state we were in, the expectation value of an operator A is: • Since we don’t know which state it is, we must weight it by the probabilities:

43. Sample Problem Prove, using thestate operator, that: for operators A that have no time dependence Sample Problem Show that is constant

44. Comments on Entropy • We showed in the previous problem that Tr(2) is constant • Easily could have generalized it to any power • Generalizes to trace of any function of  • The quantum mechanical entropy is not changed by Schrödinger’s Equation • Pure states should always evolve into pure states • It is disputed whether this applies to gravity • We don’t have a quantum theory of gravity • In principle, if you put something in a black hole, it eventually comes back out as black body radiation with a lot of entropy

45. Postulates in Terms of the State Operator • Note that to get (t), you don’t need |i(t) and fi, you just need (0) • This suggests we could write postulates in terms of the state operator • The equation above becomes our second postulate Postulate 1: The state of a quantum mechanical system at time t can be described as a positive semi-definite Hermitian operator (t) in a complex vector space with positive definite inner product Postulate 2: When you do not perform a measurement, the state operator evolves according to where H(t) is an observable.

46. Measurement Postulates for the State Operator • Postulate 3 (measurements correspond to observables) doesn’t need changing • Postulate 4 concerns the probability of getting a result a if you measure A. Postulate 4: Let {|a,n} be a complete orthonmormal basis of the observable A, with A|a,n = a|a,n, and let (t) be the state operator at time t. Then the probability of getting the result a at time t will be

47. Post-Measurement State Operator • This one is tricky • Need to figure out what the probability that it was in the state igiven that the measurement produced a • Requires a good understanding of conditional probabilities • Need to figure out the state vector if itwas in the state iafter the measurement • Then find the new state operator

48. Post-Measurement State Operator (2) • This now serves as our final postulate Postulate 5: If the results of a measurement of the observable A at time t yields the result a, the state operator immediately afterwards will be given by

49. Comments on Postulates using State Operator • The postulates in terms of the state operator are equivalent to those in terms of the state vector Pros and cons of using the state operator approach: • The irrelevant overall phase in | is cancelled out in  • The formalism simultaneously deals with both quantum and ordinary probability • You have to work with matrices (more complicated) rather than vectors • Requires greater mathematical complexity • Postulates are slightly more complicated • Whether you believe the postulates should be stated in terms of state operators or not, they are useful anyway

50. 11G. Working With the State Operator It’s Useful • The state operator can be used even if we don’t write our postulates this way • It allows us to prove powerful theorems, and simplify what would otherwise be complicated calculations • Example 1: How do you calculate scattering if the polarization of a spin-1/2 particle is random? • Example 2: How do you calculate interactions of a particle which is produced in a particular state, but at an unknown time