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

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

3. 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 ?

4. 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.

5. 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!

6. 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

7. 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!

8. 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,

9. 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 !

10. 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

11. 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

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