Quantum Mechanics and Atomic Physics Lecture 8: Scattering & Operators and Expectation Values http://www.physics.rutgers.edu/ugrad/361 Prof. Sean Oh Summary of Last Time

„ BiPBarrier Potenti il/Tlial/Tunneling

„ Case I: EV (Scattering) 0 L

„ Probability of reflection (reflection coefficient) and probability of transmission (transmission coefficient ) Barrier Potential: E>V 0 „ Case II: E>V0 (Scattering) „ This time transmission should not be a surprise, but there are other surprises ….

L

„ As in Case I, “F” (in Ψ3) is equal to 0 because if the particle goes into region (3) there is nothing to reflect it. It just keeps going. „ Now solution in regg()ion (2) is oscillatory, not exponential as in case I. Boundary Conditions Transmission coefficient

„ If you want t o compare thi s result with your b ook , k eep i n mi nd th at :

In lecture: IRdIn Reed: Resonance Scattering

„ T becomes 1 whenever k2L=nπ, because sin(nπ)=0. „ For E>V0,,gy T oscillates with energy! (For E

At certain energies the barrier is transparent to the incident mattermatter--wave.wave. So there are certain energies for which T=1 exactly! This is called resonance scattering.. Why does T oscillate with E?

„ Let’s look at the ppyyrobability density for two extreme values of T:

L L „ T oscillates because the solution in region (2) is oscillatory and dependent on k2,,p which depends on ener gy! Operators: The Hamiltonian „ Let’s consider again the 11--dim.,dim., time-time-indep.indep. S.E.:

„ The HHiltninprtramiltonian operator:: 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).

„ Ψn ::: eigenfunctions corresponding to V(x)

„ En : Energy eigenvaluescorresponding to Ψn „ So general form of eigenvalueequation can be thought of as: „ (Operator) ( Eigen funct ion))( = (Eigenva lue))( (Eigen funct ion) „ General example: Kinetic energy and potential energy operators

„ We can consider the total energy operator H op to be the sum of the individual kinetic energy

(KEop) and potential energy (PE op) operators: Eigenstate of energy „ Is the infinite square well solution an eigenstateof energy? We will use this wavefunction throughout today’s lecture Momentum operator

„ Let’s determine the operator for linear momentum Pop „ Recall a left or right propagating matter -wave: (p>0: right propagating, p<0: left propagating)

Reed is confused about the sign. Eigenstate of momentum

„ Is the infinite square well solution an eigenstate of momentum?

„ So it is not an eigenstate of momentum.

„ MidfMagnitude of momentum is pn=√(2mE n)di) and is constant, but the direction is not determined; it can be either left or right . 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 wavewave--functionsfunctions are not eigenstatesof momentum and position …. But we can evaluate something else, whether or not it ’san s an eigenstate … 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? „ Example: what is the average score on an exam?

„ Nii students get score xii , for i=1,2,3 … (i = bin in histogram)

Reed, Chapter 4

Score in this example

„ pii is the probability of a randomly chosen student to fall in bin i „ The averaggqye of a quantity x is the sum ( over all p ossible values of x) of x times the probability of having x as the value Expectation value „ expectatMore generally, if we consider position as a continuous variable and not a discrete one, we write the average value as the expectatilion value (l(also d enoted as < <>)x>): Position expectation value for infinite square well 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 th e ot her, no matter w hat 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” 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. Expectation value of Energy

„ The expectation value of the energy for the infinite sqqjguare well state n is just the eigenvalue of that state! Expectation value of p2

„ The need for this will become clear later (next time). Revisit expectation value of energy Revisit expectation value of energy Expectation value of x2

„ Again, this is something that we will find useful later.

„ Note: is not equal to 2 Summary/Announcements „ Introduced Operators and Expectation Values „ Next time: „ Uncertainty Principle ΔxΔp ≥ /2 h „ Commutators

„ HW #4 due on Monday Oct 3!