P (x) = | (x)|2 = (x) · ⇤(x)

� =h

p! p =

h

�= ~k

~ = h/2⇡ = 1.055⇥ 10�34 J s

�x�p � ~2

< K >=�2p

2m

� ~22m

d2 (x)

dx2+ U(x) (x) = E (x)

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From Reading:Lecture 15: Waves, Probability & Uncertainty 1. Waves and Probability 2. Heisenberg’s Uncertainty

Principle 3. Schrödinger’s Equation

Announcements• Virtual PS starts tomorrow. See

webpage for details. • Office hour zoom room is open

24 hours for students to work together.

• Unit 3: Check calendar pages for updated readings and homework.

• Drills are now due Weds @ 11 a.m. Reading Quizzes due Thurs @ 11 a.m.

• No Hand-in set for Monday. • This just in: all students will have

the option to take any course P/F. Talk to your adviser for details.

A physicist you should know:

Dr. Maria Goeppert-Mayer 1906-1972

Received the 1963 Nobel Prize in physics for proposing the nuclear shell model of the atomic nucleus.

CT1: Plotted at right are the wavefunctions for 3 electrons. Which wavefunction represents the electron that is moving the fastest?

1) Wavefunction A

3) Wavefunction C

5) Not enough information

2) Wavefunction B

4) Equal for all 3 functions

A

B

C

x

x

x

ψ

ψ

ψ

CT2: A particle is associated with a probability density: P(x) = 2x from x = 0 to x = 1, and P(x) = 0 for all other values of x. What is the probability that the particle would be found between x = 0 and x = 1/2

1) 0

3) 1/2

5) 1

2) 1/4

4) 3/4

6) 2

P(x)

0

0.5

1

1.5

2

x0 1

CT3: You trap a particle in a very small box, such that the spread in position σx must be quite small. What does Heisenberg’s Uncertainty Principle imply about this particle? For each of the questions below hold up: 1=true 2=false

A) A small spread in position σx implies a small spread in momentum σp (and therefore velocity).

B) There is a minimum kinetic energy that the particle must have.

C) If we make the box smaller, the minimum kinetic energy of the particle becomes smaller

L