Operators, Expectation Values and Uncertainties

Question 1

What are the Hamiltonian, Momentum and Spatial operators?

Answer 1

Question 2

To determine the effect of one operator, or multiple operators, what do you do?

Answer 2

You apply the operators to a test function f(x)

Question 3

Answer 3

Question 4

Answer 4

Lct:11 for more processed steps

Question 5

Answer 5

The eigenvalue can take ANY value for a momentum eigenstate- momentum is continuous, not quantised, unlike energy.

Question 6

INFO: Just take the solution as it is, do not need to prove that u(x) is delta function.

Answer 6

Question 7

Answer 7

Question 8

What is Postulate 2?

Answer 8

Question 9

N2K

Answer 9

Let u(x) = v(x) + w(x)

No, A is not a linear operator

Question 10

N2K What is the property of the Hermitian operator? State in Bra-ket and integral form.

Answer 10

Question 11

N2K Why must observables correspond to Hermitian Operators?

Answer 11

The eigenvalues of Hermitian operators are real, hence are observable - why they represent experimental observables.
The eigenvectors of Hermitian operators form an orthogonal basis

Question 12

Hint: Use two generic functions f(x) + g(x)

Answer 12

Question 13

Hint: Use two arbitrary functions f(x), g(x)

Answer 13

Question 14

Answer 14

Units of energy = h(bar) * omega

Question 15

State V(x) for a Quantum Harmonic Oscillator (QHO)

Answer 15

Question 16

State the ladder operators of a QHO

Answer 16

Question 17

What is the result of applying both ladder operators on each other? Rearrange for the Hamiltionian.

Answer 17

Question 18

What is Postulate 3?

Answer 18

Question 19

What does it mean if PSI(x) is already an eigenstate of the Operator?

Answer 19

It means EVERY MEASUREMENT is known with complete certainty (does NOT CHANGE- SAME EIGENVALUE AND EIGENVECTOR).

This is called a determinant state

Question 20

IMPORTANT: u2(x) is an eigenstate of the Hamiltonian, hence it is a determinant state, and all measurements will return the same value.

Question 21

What happens if PSI(x) is not an eigenstate(i.e. a superposition of eigenstates)?

Answer 21

PSI(x) is not a determinant state, hence we can no know with certainty that every measurement will return the same value. It is likely a different eigenstate of the superposition (can only be the eigenstates of the superposition - all other amplitudes are 0) will be measured.

Question 22

State the probability of obtaining the nth eigenstate upon measuring the observable of a superposition state. (Born’s Rule)

Answer 22

Question 23

Answer 23

Normalise is the first step

Question 24

Important: Measuring observables with two operators of the SAME basis return eigenvalues(different) corresponding to the SAME eigenvector- will always measure same eigenstate.

The system is in a deteminant state. When the same eigenstate is returned for two operators, they are compatible.

Answer 24

Question 25

Answer 25

Question 26

Important: Upon change of basis, all previous information on the state is lost, hence eigenstates can change.

Answer 26

Question 27

State the commutation relation for compatible operators B and A.

Answer 27

Tells us associated observables are compatible

Question 28

Answer 28

Question 29

In making a measurement we lose all information about the orginal state (as long as Psi in not already in an eigenstate i.e. is a superposition state). How do we get a picture of the origininal state? Take repeated measurements! What is the average of all measurements called?

Answer 29

The average of a set of repeated measurements of a Quantum state, not in an eigenstate is called the EXPECTATION VALUE. The image is more relevent to building up the orginal quantum state by taking many, many measurements.

Question 30

What is the equation for the expectation value in Bra-ket form?

Answer 30

Question 31

What is the expectation value equal to? (Sum form- find by using bra-ket notation with the sum notation of a quantum state) What does this sum represent?

Answer 31

The sum represents the average measurement O n weighted by its respective probability (P = Cn **2)

Question 32

Answer 32

You can use the known result for the Expectation value as shown, using cn**2 and On (the eigenvalue which is in this case = n) or do it the longer way round starting with which generates the same results.

Question 33

Answer 33

Question 34

What is the uncertainty of an operator? (Give the equation of uncertainty)

Answer 34

Question 35

Hint: Consider the state Iv>

Answer 35

Question 36

What would a series of measurements of a sytem in an eigenstate be equal to? What is its uncertainty?

Answer 36

Question 37

What is the uncertainty for the following system not in an Eigenstate?

Answer 37

Question 38

Question 39

State the generalised uncertainty principle

Answer 39

Question 40

Using the genralised uncertainty principle derive Heisenberg's Uncertainty Principle

Answer 40

Question 41

What is Postulate 4? Hint: Time dependence and evolution of a system.

Answer 41

Question 42

Answer 42

Question 43

What is the expression for the wavefunction of dependence in summation notation? Hint: The 'weighting' does change with time.

Answer 43

Since I un > has no explicit time dependence, Cn must be time dependent --> Cn(t) for PSI(x, t) to have explicit time dependence. un(x) is a time independent function/ket.

Question 44

Use the Postulate 4 to derive the full quantum state of the system. Hint: First find Cn(t) in terms of Cn(0), e^f(t). Postulate 4 in image if you do not remember.

Answer 44

Question 45

What is the TISE in ket notation?

Answer 45

Question 46

What is Cn(t)? Use Cn(t) to write the full quantum state in Ket notation.

Answer 46

Question 47

What is the full time-dependent quantum state for any Quantum State?

Answer 47

Question 48

Show that the epectation for the Hamiltonian is not time-dependent by acting on an eigenstate Iun>. Hint: For an eigenstate, Cn(0) = 1

Answer 48

- Write in time dependent form. - Find . The expectation value for the Hamiltonian is time independent, even though the state is.

Question 49

Show that the probabilty density of an energy eigenstate remains constant.

Answer 49

ONLY APPLIES TO ENERGY EIGENSTATES. Probability densities remain fixed in time.

Question 50

Derive the expectation value of the Hamiltonian for time dependent superposition states. Compare the result to the expectation value of the Hamiltonian of superposition states with no time dependence. Start with psi given in image.

Answer 50

Time dependence does not effect the Hamiltonian. The expectation value of the Hamiltonian is totally time independent and has 0 time evolution. Same result.

Question 51

Which expectation values are time independent for stationary states?

Answer 51

Position, Momentum and Hamiltonian expectation values are time independent for stationary states.

Question 53

What is the Bra of Postulate 4? When is this used?

Answer 53

This is used in finding if an operator/measurement has time dependence.

Question 54

Test whether an arbitrary Operator, O's expectation value is time dependent. Hint: Expectation, d/dt, 4th postulate.

Answer 54

Question 55

Find the Ehrenfrest theorem for a general operator O. You may find the image attached useful.

Answer 55

Question 56

When is the expectation value of an operator time independent?

Answer 56