1 Reading: QM course packet Ch 5 up to 5.6 SUPERPOSITION, MEASUREMENT, NORMALIZATION, EXPECTATION VALUES

2 n =1,2,3,4,5… for x a

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4 Quantum Measurement: In the spins paradigm, you became familiar with the idea that the state of a spin-1/2 particle might be a superposition of eigenstates up and down, rather than a pure eigenstate, and one can talk only about the probability of measuring the z-component of spin angular momentum as up or down. In the context of wave function language, and using the particular example of the finite potential energy well, we can ask similar questions … Given a particular wave function (state) of a particle, what is the probability that a particular energy will be measured for the particle? A particular momentum? And so on. The key is superposition.

5 The quantum state is a superposition of (energy) eigenstates. The result of an (energy) measurement may be ONLY an eigenvalue of the (energy) operator. The coefficients of the eigenstates allow us to infer the probability that a particular eigenvalue will be measured. Do we all measure the same energy? What information do we have about the quantum state? How can we infer what the quantum state is, and conversely, given a particular quantum state, can we make predictions about what we might measure in a given measurement?

6 The quantum state is a superposition of eigenstates. general stateeigenstatecoefficient

7 Initial state Suppose the result of single energy measurement is E 2 State after single energy measurement Probability that the result of single energy measurement is E 2

8 The act of measurement has to do with projection. Projection of a general state onto an eigenstate of the quantity being measured yields the AMPLITUDE coefficient. The PROBABILITY that the result of a particular measurement yields E 2 is given by the modulus SQUARED of the amplitude coefficient or projection: Class to show that: This is because eigenstates of the Hamiltonian (energy) operator are orthogonal (also true for any other Hermitian operator):

9 The probability that a measurement on a state will yield SOME EIGENSTATE must be unity, so What if Same RELATIVE probabilities. Always possible to scale all coefficient values to make sum of "squares" =1 ("normalizing" the coeffs). Discuss:

10 Orthogonality and projections using functions Continuum analog

11 Abstract bra-ket & position state (wavefunction) representations State superposition Eigenstate orthogonality Normalization Expectation value Eigenstates & eigenvalues

12 Class to show that in general Measurement of a quantity Q (maybe energy, maybe momentum….), represented by an operator, may yield values q 1, q 2 … etc. (suppose we know the q i ). The AVERAGE or EXPECTATION value is denoted by Discuss :

13 Example where Q is the Hamiltonian or energy operator, and we expand quantum state in basis of eigenstates of H. Bra-ket notation: Is  an eigenstate of the Hamiltonian or energy operator? NO !!

14 Example where Q is the Hamiltonian or energy operator, and we expand quantum state in basis of eigenstates of H. Bra-ket notation

15 Is  (x) an eigenstate of the Hamiltonian or energy operator? Example where Q is the Hamiltonian or energy operator, and we expand quantum state in basis of eigenstates of H. Wave function representation NO !!

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18 Example where Q is NOT the Hamiltonian or energy operator, and we expand quantum state in basis of eigenstates of H (and they are NOT eigenstates of the Q operator). Wave function representation: Well, now there are fewer shortcuts. You have to put in the specific wave functions, put in the specific operator, and evaluate an integral. Sometimes, there are symmetry arguments that help you evaluate the integral.

19 So If general state is normalized:

20 Coefficients: the modulus squared (probability of measurement) is a weighting factor to determine average. Expectation value of energy:

21 Coefficients: They are the projection of the general state onto the eigenstate. Do specific example on board Projections: Remember Fourier coefficients?

22 Eqn 2.82 and 2.83 in QM text – go over this. Examples from infinite well Uncertainty: