Summary of Last Time Barrier Potential/Tunneling Case I: E<V 0 Describes alpha-decay (Details are in the lecture note; go over it yourself!!) Case II:

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1 Quantum Mechanics and Atomic Physics Lecture 8: Scattering & Operators and Expectation Values Prof. Sean Oh

2 Summary of Last Time Barrier Potential/Tunneling Case I: E<V 0 Describes alpha-decay (Details are in the lecture note; go over it yourself!!) Case II: E>V 0 (Scattering) L Probability of reflection (reflection coefficient) ) and probability of transmission (transmission coefficient)

3 Barrier Potential: E>V 0 Case II: E>V 0 (Scattering) This time transmission should not be a surprise, but there are other surprises. L As in Case I, F (in Ψ 3 ) is equal to 0 because if the particle goes into region (3) there is nothing to reflect it. It just keeps going.

4 Now solution in region (2) is oscillatory, not exponential as in case I.

5 Boundary Conditions

6 Transmission coefficient If you want to compare this result with your book, keep in mind that: t In Reed: In lecture:

7 Resonance Scattering T becomes 1 whenever k 2 L=nπ, because sin(nπ)=0. For E>V 0 0, T oscillates with energy! (For E<V 0, T increases with energy, as expected) At certain energies the barrier is transparent to the incident matter-wave. So there are certain energies for which T=1 exactly! This is called resonance scattering.

8 Why does T oscillate with E? Let s look at the probability density for two extreme values of T: L T oscillates because the solution in region (2) is oscillatory and dependent on k 2 2, which depends on energy! L

9 Operators: The Hamiltonian Let s consider again the 1-dim., time-indep. indep. S.E.: The Hamiltonian n operator: r

10 Operators and general form of eigenvalue equation An operator does something to a function and returns a result. Recall: only certain function Ψ(x) will satisfy S.E. for a given V(x). Ψ n : eigenfunctions corresponding to V(x) E n : Energy eigenvalues corresponding to Ψ n So general form of eigenvalue equation can be thought of as: (Operator) (Eigenfunction) = (Eigenvalue) (Eigenfunction) General example:

11 Kinetic energy and potential energy operators We can consider the total energy operator H op to be the sum of the individual kinetic energy (KE op ) and potential energy (PE op ) operators:

12 Eigenstate of energy Is the infinite square well solution an eigenstate of energy? We will use this wavefunction throughout today s lecture

13 Momentum operator Let s determine the operator for linear momentum P op Recall a left or right propagating matter-wave: (p>0: right propagating, p<0: left propagating) Reed is confused about the sign.

14 Eigenstate of momentum Is the infinite square well solution an eigenstate of momentum? So it is not an eigenstate of momentum. Magnitude of momentum is p n = (2mE n ) and is constant, but the direction is not determined; it can be either left or right.

15 Eigenstate of position Is the infinite square well solution an eigenstate of position? So it is not an eigenstate of position. We find that the infinite square well wave-functions are not eigenstates of momentum and position. But we can evaluate something else, whether or not it san eigenstate

16 Useful operators Reed, Chapter 4

Average value Question: what is the position of the particle? Answer: given in terms of a probability distribution A more meaningful question: what is the average position of the particle?

17 Average value Question: what is the position of the particle? Answer: given in terms of a probability distribution A more meaningful question: what is the average position of the particle? Example: what is the average score on an exam? N i students get score x i, for i=1,2,3 (i = bin in histogram) Reed, Chapter 4 Score in this example p i is the probability of a randomly chosen student to fall in bin i The average of a quantity x is the sum (over all possible values of x) of x times the probability of having x as the value

18 Expectation value More generally, if we consider position as a continuous variable and not a discrete one, we write the average value as the expectation value (l (also denoted d as <x>): <>)

19 Position expectation value for infinite square well

20 Position expectation value for infinite square well This result means that average of many measurements of the position would be at x=l/2. It is independent of n! Well is symmetric, so particle does not prefer one side of well to the other, no matter what state n it is in. Note that Ψ is sometimes zero at x=l/2! So expectation value means average value not most probable value

21 Momentum expectation value for infinite square well See appendix C in Reed for useful integrals Again is not surprising. The well is symmetric so the particle should have no preference for traveling one way or the other.

22 Expectation value of Energy The expectation value of the energy for the infinite square well state n is just the eigenvalue of that state!

23 Expectation value of p 2 The need for this will become clear later (next time).

24 Revisit expectation value of energy

25 Revisit expectation value of energy

26 Expectation value of x 2 Again, this is something that we will find useful later. Note: <x 2 > is not equal to <x> 2

27 Summary/Announcements Introduced Operators and Expectation Values Next time: Uncertainty Principle ΔxΔp h/2 Commutators HW #4 due on Monday Oct 3!

Momentum expectation Momentum expectation value value for for infinite square well

Momentum expectation Momentum expectation value value for for infinite square well Quantum Mechanics and Atomic Physics Lecture 9: The Uncertainty Principle and Commutators http://www.physics.rutgers.edu/ugrad/361 Prof. Sean Oh Announcement Quiz in next class (Oct. 5): will cover Reed