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Scattering/Operators and Expectation Values 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: E<V0 Describes alpha -decay ( Details are in the lecture note; go over it yourself!!) Case II: E>V (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 wan t 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<V0, T increases with energy, as expected) 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., timetime--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 eigenstate of 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 More 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 o f we ll to t he 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: <x2> is not equal to <x>2 Summary/Announcements Introduced Operators and Expectation Values Next time: Uncertainty Principle ΔxΔp ≥ /2 h Commutators HW #4 due on Monday Oct 3!.
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