Week 3 Quantum

Question 1

Why is the wave function not a de broglie particle? (5 points)

Answer 1

• Can be complex
• Can be infinitive
• Hamilton can be infinite until x goes to infinity (tend to measure infinite particles)
• One wave function can describe many particles
• Exists in space with more than 3 dimensions (space and time)

Question 2

Born rule

Answer 2

Psi squared - probability of finding a particle at position x

Question 3

Wavefunction provides

Answer 3

All information required to make predictions about system

Question 4

Can you measure wavefunctions?

Answer 4

No because can only measure real numbers

Question 5

Infinite square well wavefunction

Answer 5

B cos (pi x/ 2a)

Question 6

Infinite square well wavefunction

Answer 6

B cos (pi x/ 2a)

Question 7

Mean position

Answer 7

Integral of x * psi^2

Question 8

TIWE with expansion coefficient

Answer 8

Sum of Cn Psi n

Question 9

TDWE with expansion coefficient

Answer 9

Sum of Cn Psi n e^ -(i En t/ h bar)

Question 10

Wave function Psi n x Psi m

Answer 10

0 if n does not = m, 1 if n=m due to orthognality

Question 11

Physical significance of expansion coefficients

Answer 11

Cn squaed =probability of measuring En

Question 12

Cn

Answer 12

Expansion coefficient = integral of Psi x Psi*

Question 13

Sum of Cn squared

Answer 13

=1

Question 14

Eigenfunction

Answer 14

Wavefunction defining state where observable has known, defined value

Question 15

Eigenvalue

Answer 15

Value of observable physical parameter in eigenstate

Question 16

Eigenstate

Answer 16

State of a system where observable has known value

Question 17

Probability of being in ground state

Answer 17

Probability of being in C1 = C1 squared

Question 18

Find all Cn (7 steps)

Answer 18

1. Find any constants in wave function E.g. A
2. Now define Wavefunction (WF)
3. Cn = Psi n x WF
4. Divide into Psi odd and Psi even (odd cos, sin even)
5. Cn odd = integral of WF x (1/sqrt a) cos (n pi x/ 2a)
6. Cn even = integral of WF x (1/sqrt a) sin (n pi x/2a)
7. Therefore Psi = Cn written in individual surviving terms…

Question 19

Momentum operator

Answer 19

Px = - i hbar d/dx

Question 20

The eigenfunctions depend on…

Answer 20

The observables. There are different eigenfunctions for each observable but all included in wave function

Question 21

Discrete vs continuous variables…

Answer 21

Exponential: Discrete =n, continuous = k

Question 22

Operator for position

Answer 22

X

Question 23

How to find E1 - two options

Answer 23

1. Hamilton operator on Wavefunction
2. = whatever is in front of integral of Wavefunction squared

OR c1 squared

Question 24

Expectation value of ISqW

Answer 24

Integral of position, operator x psi squared

Question 25

Wavefunction of ISqW

Answer 25

1/sqrt a cos (pix/2a)

Question 26

Find sigma

Answer 26

1. Standard deviation 2. = Sqrt ( - 2>^2)

Question 27

Operators are…

Answer 27

Objects that act on wavefunctions to return another wave function

Question 28

How do you find answer if normalised

Answer 28

Integrate and make =1 Write answer In terms of A =

Question 29

Integral of (b - x) ^2

Answer 29

- (b - x) ^3

Question 30

If exponential has absolute value of x…

Answer 30

1. Need two integrals (x and -x) 2. Must be even function

Question 31

Ehrenfests theorem

Answer 31

Expectation values of displacement and momentum obey time evolution equations analogous to classical mechanics

Question 32

If I measure the energy of the particles, what value do I obtain?

Answer 32

En, with probability (magnitude of Cn)^2

Question 33

What is wave function collapse?

Answer 33

Going from an initial wave function with the possibility to measure a whole range of energies to a single eigenfunction with a specific energy eigenvalue

Question 34

What is the measurement problem?

Answer 34

We can predict the probability of measuring a certain value, but never know with certainty what the exact result will be, (unless we know the particle is in an eigenstate).

Question 35

What is the Copenhagen/orthodox interpretation?

Answer 35

There is a quantum world which is abstract containing wavefunctions and objects without defined properties, and then our world. The worlds are joined by measurement.

Question 36

What is the Bohmian interpretation

Answer 36

The universe only appears probabilistic the wavefunction is a pilot wave, guiding real particles with real properties

Question 37

What is the many worlds interpretation

Answer 37

The universe has a function and describes. Collapse never happens. All eigenstates exist simultaneously.