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Postulates of QM Part 1: Comparing Classical and Quantum Mechanics Part 2: Complications and Uncertainty Part 3: Heisenberg Picture and DOF's Why Quantum Mechanics? Catastrophic Failures of Classical Physics Radical sameness of atom, etc. Spectral lines, lifetimes Ex: Barium atoms made in a nuclear reactor and Barium atoms left over from the earliest stars are exactly the same. Why Quantum Mechanics? Catastrophic Failures of Classical Physics Radical sameness of atom, etc. : Spectral lines, lifetimes Observation of matter waves: Davison and Germer Davison and Germer shined an electron beam on cleaned crystaline Nickel and found that the electrons came off at the “Bragg” angle, rather as if they were a form of x-rays. This was in accordance with the deBroglie relation, Why Quantum Mechanics? Catastrophic Failures of Classical Physics Radical sameness of atom, etc. : Spectral lines, lifetimes Observation of matter waves: Davison and Germer Quantities exist without any classical counterpart: Spin Spin is like an internal angular momentum that particles possess. It is not actually angular momentum in that no spatial version of spin exists as far as we know, whereas angular momentum will be associated with spatial properties of the particles' wave. Comparison of QM and Classical Physics Comparison of QM and Classical Physics Classical QM Condition of system Comparison of QM and Classical Physics Classical QM Condition of system “The State” Hilbert (complex vector) Space QM is a temporal succession of vectors in this vector space. Phase Space Classical is the motion of the x,p(t) point in States can be: 1) States representing a single thing: for example a single barium atom. = |Barium> 2) Or, any superposition (that is, a linear combination) of other states; = a|Barium> + b|Ytterbium> Note 1: 'a' and 'b' are in general complex numbers Note 2: The fact that a quantum theory must have facility with superpositions is the reason that that linear algebra studied in Chapter 1 is going to be so useful. Comparison of QM and Classical Physics Classical QM Condition of system Observables/measurement eigenvalues of , a linear operator Comparison of QM and Classical Physics Classical QM Condition of system Observables/measurement eigenvalues of , a linear operator NOTE: A list...(could be discrete) Generally, Continuous Functions Comparison of QM and Classical Physics Classical QM Condition of system eigenvalues of Observables , a linear operator Reality guarantor of real eigenvalues... i.e. Hermiticity of the operators associated with measurable quantities is necessary because we only measure real quantities...no gauges or meters in the lab give complex numbers directly! Measurement and Observables Throughout the following pages let the be the orthonormal basis of eigenvectors of a hermitian operator with A measurement in quantum mechanics is both (1) (An Activity: ) a projection into the eigenspace of the corresponding eigenvalue measured. (2) (Probabilistic:) the frequency of measuring a particular eigenvalue is proportional to the square of the overlap of the given state with the corresponding eigenspace. Measurement and Observables Ex 0: Suppose we start with a pure state w.r.t. = Then, the repeated measurement of always yield the number will remain while in this state will and the state vector Ex 1: Mixed state case: the non-degenerate case Suppose : The a single, isolated measurement of will return just one value of three possible numbers , , or ONLY ! But you can't know which value until you “measure” it. We re-iterate, that you will never find an intermediate value of, say, ( + )/2 .Essentially, from a single isolated measurement of this is atomism. In words, no matter how the system was prepared (how mixed), when you perform a measurement you will always measure a discrete value that is an eigenvalue of the observable. You can have one Barium atom. Or one Yterbium atom. Your state can be an admixture of the two, but it is not real to find for a single measurement an atom that is some combination of the two. ...So if it was a mixed state of a Barium and Yterbium atom...and you measured it to be Barium...then what? In quantum mechanics, as we are teaching it here, a single measurement actively places the state in the eigenbasis corresponding to the eigenvalue measured. For our example, suppose we start with We perform a single measurement and find Then, were we to immediately repeat this measurement, each subsequent measurement would yield the same value, ...The Barium atom would remain the Barium atom.... BUT: Then what does it really mean to be in a mixed state? Probability and Quantum Mechanics “The Copenhagen Interpretation” The probability of a particular measurement outcome is proportional to the norm square of the state's overlap with the associated eigenbasis. = and (Important note: this formula assumes that both are normalized.) So, the admixture coefficients reflect the likelihood of the outcomes of any particular measurement...said another way, the frequency of a particular measurement outcome from many independent measurements on the (each time identically prepared) state. Ex 2: = ¼ , ¼ , ½ for the outcomes , respectively. , 'Collapse of the State Vector' But back to that atomisim...if we make a measurement on an (arbitrary) state vector and find a value for example, we expect each immediate re-measurement of that same observable to again give But this means that the subsequent probability of measuring the observable and finding is one. That in turn by the Copenhagen interpretation means that as a result of the measurement process itself there can no longer be any superposition of states with different eigenvalues...for our example, the state would have to be a pure state. = The formerly mixed state has been projected onto a pure state by the activity of measurement... This is the so-called 'collapse of the state vector'...in the sense that the initial mixed state has 'collapsed' onto an eigenstate. Thus all the information about the mixture of states in the state vector before the measurement has been completely obliterated by the measurement process (so defined). ...whether the atom was produced in a nuclear reactor a microsecond ago or was left over from the earliest stars, if we measure them both to be Barium atoms they are radically identical. When identical measurements could lead to different states “The Degenerate Case” Suppose the state vector was Where, we have two different states with the same , eigenvalue “Degenerate States” Assume that they are orthonormal -WOLOG- When identical measurements could lead to different states “The Degenerate Case” Suppose the state vector was Where, we have two different states with the same , eigenvalue NOW... Suppose that we make a measurement of and find the value the measurement? . What is the state of the system after When identical measurements could lead to different states “The Degenerate Case” Suppose the state vector was Where, we have two different states with the same , eigenvalue NOW... Suppose that we make a measurement of and find the value the measurement? ANS: . What is the state of the system after Averages By the Copenhagen interpretation, the average value of the measurement of state is given by while the system is in = This is called the Expectation Value of Ex 1: For The expectation value is =¼ +¼ + ½ in Ex 2: If the state were be? What would ANS: = ¼ +3/4 NOTE 1: These expectation values are average values, and as such can take on continuous sets of values, unlike individual quantum measurements. NOTE 2: Physical examples of inescapably average values in quantum mechanics are things like the lifetime of an excited state. Lifetime has no real meaning as an individual measurement, but it does as a (generalization of) an expectation value. Uncertainty Uncertainty denotes a measure of the spread of the individual measurement of an observable. It is therefore a state-dependent notion. A useful mathematical definition is : NOTE: This uncertainty is positive, real. NOTE: It is the same as the notion of standard deviation in which you use as the distribution. Uncertainty: an example Given Compute the uncertainty on of on this state. Uncertainty: an example Given Compute the uncertainty on of on this state. ANS: Recall the goal is: And recall from the previous page that +¼ =¼ SO; only need to figure out As an operator, since = + ½ expectation value And, applying 2 to this gives, = So we are now ready to assemble these pieces together into a measurement of the uncertainty; ]2 = [ = [ ]2= 2 2 - - Non-commuting bases of measurement Take two hermitian operators . The commutator Then it is possible to find a basis which Iff diagonalizes Iff and and (the “compatible operators” case) is not zero, then in general the operators cannot be simultaneously diagonal, called then “incompatible operators”. Non-commuting bases of measurement Take two hermitian operators and Let: And take as a starting state the vector In Pictures ! . The commutator Suppose we measure To find value a1 The process of measurement has projected our system into state |t2>. We now measure This is the case where is not zero... Then, suppose one measures b2 Then we have Note this is not the same as first measuring And then measuring to get a1 to get b2 And finally, ... If the observables are compatible... =0 So they can be simultaneously diagonalized... Now perform measurements on |t1> After this, one will never get b2...only b1 Summary about measurement in QM a) Non-commutative joint probability: Let and have eigenvalues respectively. Let Let and Be the respective probabilities in a single measurement. . be the joint probability of measuring and then immediately measuring Then: In general note that: (incompatible case) iff (compatible case) Then Note: There is really no way to reduce the probabilistic nature of quantum reality to probability functions (strictly positive, single valued densities) on classical phase space. Comparison of QM and Classical Physics Classical QM Condition of system Observables Reality Determinism Quantum Determinism: at time t=0 Given the state Specified with complete precision one can find the state and the complete Hamiltonian, At any subsequent time with no uncertainty. Classical Determinism: Given the position(s) and momenta at time t=0 with complete precision, and the complete Hamilton, the subsequent position(s) and momenta are then known at any subsequent time with no uncertainty. Quantum Determinism: at time t=0 Given the state Specified with complete precision one can find the state and the complete Hamiltonian, At any subsequent time with no uncertainty. Classical Determinism: Given the position(s) and momenta at time t=0 with complete precision, and the complete Hamilton, the subsequent position(s) and momenta are then known at any subsequent time with no uncertainty. UPSHOT: Both QM and Classical are causal theories. All the 'probability/uncertainty' in QM comes from the measurement 'process'. Heisenberg Uncertainty Principle ( (See Shankar, Chapter 9) Example Translation from QM to Classical Classical QM x p ) Canonical Commutation Relation Poisson Bracket Complication: Operator Ordering... Differential relationship x and p are just numbers.... Operator Ordering So, the Universe is bumping and grinding away, rotating its state vector...and we want to relate combinations of operations on that state vector as things that make classical sense to us, for example, angular momemtum or some kind of perturbation. How do we correspond (combinations of) operators acting on the state vector with classical notions? There is in general no unique way to translate backwards from a classical notion to a quantum notion! But we can try... Example: Kinda like angular momentum.... This operator is fine in the quantum theory. It does not however represent the classical quantity 'xp'. For one thing, the classical quantity is always real, whereas this quantity is not Hermitean and so does not always have real eigenvalues..... Operator Ordering To make Hermitean and thus have real eigenvalues one can try the following combinations of related operators 1 2 Q: Are both Hermitean? Q: Why is only one of these choices related to the classical observable xp? Which one? Example Translation from QM to Classical Classical QM x ) Canonical Commutation Relation p Poisson Bracket In the oft-used position basis above, these are equivalent to: Example Translation from QM to Classical Classical QM x ) Canonical Commutation Relation p Poisson Bracket In the oft-used position basis above, these are equivalent to: But not every observable has a classical version... Example Translation from QM to Classical Classical QM x ) Canonical Commutation Relation p Poisson Bracket Spin ? The Schroedinger Equation Setting up the equation: Find an operator realization the that captures the physical details of the system. Often this can be done by promoting the co-ordinates and momenta to operators as we described earlier in this talk. The multi-dimensional case: The maximal subalgebra of all operators that among themselves commute with each other is called the “Maximal Set of Commuting Observables” or also, the Cartan subalgebra (CSA). Since they commute with each other they are compatible. That in turn means that we can classify all states of the system in terms of eigenvalues with respect to each operator in the CSA An example of this is related to the classification of states in problems involving (a) Spatial dimension (b) Multi-particle systems. Ex: Co-ordinates in 3-d commute! THUS, the eigenvalues of the position operators must form a good basis for the Hilbert space. Now these operators, being positions, have a continious spectrum. Thus that Hilbert space describing a single particle in 3-d is simply the space of all function depending on three (position eigenvalue) co-ordinates. Ex: -> The Harmonic Oscillator in 3-d (isotropic case) Classical Hamiltonian = Descends rather simply from replacing the operators in this Hamiltonian below with their classical counterparts. = We'll study the Scroedinger equation that is associated with this Hamiltonian in somewhat more detail later...for now we note only that in the co-ordinate basis this differential operator can be written in different co-ordinate frames. To do that, go to the co-ordinate basis of the operators; And so the Hamiltonian becomes; = Note the appearance of the laplacian operator, Which you have already studied in your E&M class. You know how to transform it to other co-ordinate systems, so for example, in spherical co-ordinates, one has; = +r2 Multi-particle case Q: How do we use Quantum Mechanics to talk about multiparticle systems? A: The operators associated with the positions of all the particles are expected to commute with each other; Think of the physical meaning of the position eigenbasis... So, in the position eigenbasis, our Hilbert space is the space of all functions of N variables...where N is the number of particles times the spatial dimension! Ex: to follow later.... The Schroedinger Equation Objective : Given Solve To find: EX: Time Independent Step1: Find energy eigenbasis: with Step 2: Study the temporal evolution of the energy eigenbasis: Let : Then, the Schroedinger equation become an ordinary DE for the coefficients: Which can be solved as So that the energy eigenbasis in time is = Step 3: Expand the solution in terms of the energy eigenbasis ..and these coefficients must solve the same equation as before ! Thus, Where now, evaluating both sides at t=0 gives, That's it ! Pretty simple! SO the general solution is; Example of using this solution method: Quantum Beats Physicist JJ studies the wavevector of the object of his affection, called which lives in a two-dimensional vector space. The affection operator is an observable which has two eigenvectors, called |SheLovesMe> (or |SLM> for short) and |SheLovesMeNot> (|SLMN> for short), of eigenvalues +1 and 0 respectively. |SLM> = |SLM> |SLMN> = 0 Show: if an operator has eigenvalues +1 or 0 only it must be a projection. Note: = so is a projection Example of using this solution method: Quantum Beats JJ wants to cruise through time with the object of his affection. Fortunately, he knows the Hamiltonian that evolves its wavefunction so he can track it and get an idea of what he can expect of measuring affection as time passes. The Hamiltonian is and further, let its eigenvalues be and Case 1: If they commute, SHOW: The |SLM> and |SLMN> must be eig-vects of (wolog) |SLM> = E1 |SLM> |SLMN> = E0 |SLMN> Example of using this solution method: Quantum Beats Case 1: (con't) SO, If at time t=0, JJ measures State of the system at t=0 is then to find 1. The =|SLM> Then use But since |SLM> is an energy eigenstate, this is solved by Which means that JJ will always measure Show: =0 for this state of affairs... to be 1. BUT... Example of using this solution method: Quantum Beats BUT: actually time and affection are not so kind to JJ... General Case: Meaning that the eigenbasis of say, |1> and |0>, are different than the eignebasis of For definitness, take the bases to be 450 apart, so that Example of using this solution method: Quantum Beats General Case:(con't) So now, he again measures affection at time t=0 to find one ( so But, converting that to H-eigenvectors and evolving them gives, are the energies of the states in frequency units. This solution means that after a certain time the two will accumulate enough relative phase components of so that the state will lie along the |SLMN> direction ! Where So measuring will give time dependent results! Example of using this solution method: Quantum Beats General Case:(con't) So, as one measure of the change, lets compute the expectation value of in time. In steps, first note: So that we get: Stop & Think: What happens to this in the w1-w0 limit and why? Example of using this solution method: Quantum Beats General Case:(con't) So So, the time-averaged value of the measurements is 1/2 It clearly can be 0, in which case one must conclude that we are in the |SLMN> state. Operationally, the measurement of this observable oscillates in time...a “quantum beat” at the difference frequency of the states. Example of using this solution method: Quantum Beats General Case:(Class Discussion) Are there measurement protocols that guarantee happier outcomes for JJ? How can he pick the petals for more 1's? Uncertainty of Let us now compute the uncertainty in the measurement of this observable. Since L is a projection, we simplify our uncertainty formula to which becomes, So that the time average of the uncertainty is ... General formula for the solution in terms of a time-ordered product: The Propagator = = U is called the propagator, since it is the matrix that gives the state at time t given just the state at time 0 Note that it is a unitary matrix: In the general case the Hamiltonian at one time does not commute with the Hamiltonian at another time! We write the rather formal expression for the propagator ; 'T” means the time ordered product, by which we mean; = Where