Measurement and Expectation Values 2016-2 QUANTUM MECHANICS FOR ELECTRICAL AND ELECTRONIC ENGINEERS Measurement and Expectation Values Min Soo Bae 2016.10.26 mandoo113@yonsei.ac.kr School of Electrical and Electronic Engineering, Yonsei University, Korea Semiconductor Engineering Laboratory

Outline Introduction a. Quantum mechanical measurement b. Expectation values and operators c. Time evolution and the Hamiltonian Summary

E ? Introduction Introduction We have learned… Single energy eigenstate in infinite potential well We are going to learn… What is the energy for oscillator or particle in a superposition of energy eigenstates? E ?

a. Quantum mechanical measurement Probabilities and expansion coefficients Normalized quantum mechanical wavefunction Measurement postulate On measurement of a state, the system collapses into the nth eigenstate of the quantity being measured with probability When we make a measurement on a quantum mechanical system, On measurement, the system collapses into some eigenstate, maybe the nth eigenstate As far as I understand, at the moment of the measurement, the system collapses into some state, here nth state for some reason, And then the probability of the eigenstate is the modulus c sub n squared. This is quite odd and strange, but this is postulate, and not going to be explained kindly. But in the Youtube lecture, the speaker says that there are 3 reasons why we don’t explain this first, it’s best to leave this until we understand more about how quantum mechanics works. Second, from a practical point of view, this Born’s rule just turned out to work very well empirically. Third, this collapse phenomenon is that we actually don’t really know the answer.

a. Quantum mechanical measurement Expectation value of the energy Experiment to measure the energy E - repeat many times and gets a statistical distribution - get an average answer Expectation value of the energy Example – coherent state of harmonic oscillator From the previous lecture… the coherent state for a harmonic oscillator of frequency ω is where Expectation value! Not an eigenvalue!

a. Quantum mechanical measurement Stern-Gerlach experiment Non-uniform magnetic field a vertical magnet : deflected up or down a horizontal magnet : not be deflected magnet of other orientation : deflected by intermediate amounts

a. Quantum mechanical measurement Stern-Gerlach experiment with electrons Electrons – quantum mechanical property called spin magnetic moment like a small magnet e This is very odd when we think classically This experiment fits in what we have postulated. On measurement of a state, the system collapses into the nth eigenstate of the quantity being measured with probability Measuring the vertical component of the spin… two eigenstates (spin up & down) We are measuring spin along some axis!

b. Expectation values and operators Hamiltonian operator Operator : an entity that turns one function into another Hamiltonian operator is related to the total energy of the system! Define the entity From the time-independent and time-dependent Schrödinger’s equation Definition of operator is an entity that turns one function into another As you might predict from the name of operator,

b. Expectation values and operators Operators and expectation values Let’s look at this Integration. Using the definition of the hamiltonian operator inside the bracket, and this is usual expansion here Hamiltonian operator, wavefunction, expectation value of the energy We do not have to solve for the eigenfunctions of the operator to get the result !

c. Time evolution and the Hamiltonian Schrödinger’s time dependent equation : Rewriting as : Presuming V(r) is constant, And if can be replaced by a constant number, we can integrate to get Definition of operator is an entity that turns one function into another As you might predict from the name of operator, If it is legal assumption, we would have an operator that gives us the state at time t1 directly from that at time t0

c. Time evolution and the Hamiltonian Let’s think about the legality … Since is a linear operator, for any constant a and Hence, Suppose the wavefunction at time t0 is Multiplying by the time evolution factor applying power series : We can apply this to all higher powers so H hat to the power m operaing on psi sub n of r equals to E sub n to the m times psi sub n of r This time evolution operator operates on the wave function and tell us that what the state is at time t1 Plus, this is reversible, so we would know the past from the present Time evolution operator

Summary Expectation value of the energy Hamiltonian operator Operators and expectation values Time evolution and the Hamiltonian