Quantum Mechanics and Quantum Mechanics and Atomic PhysicsAtomic Physics

Lecture 8:Lecture 8:Lecture 8:Lecture 8:

ScatteringScattering

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Operators and Expectation ValuesOperators and Expectation Values

http://www.physics.rutgers.edu/ugrad/361http://www.physics.rutgers.edu/ugrad/361http://www.physics.rutgers.edu/ugrad/361http://www.physics.rutgers.edu/ugrad/361

Prof. Sean OhProf. Sean Oh

Summary of Last TimeSummary of Last TimeSummary of Last TimeSummary of Last Time

B i P i l/T liB i P i l/T liBarrier Potential/TunnelingBarrier Potential/TunnelingCase I: E<VCase I: E<V00

Describes alphaDescribes alpha--decay (decay (Details are in theDetails are in theDescribes alphaDescribes alpha decay (decay (Details are in the Details are in the lecture note; go over it yourself!!lecture note; go over it yourself!!))

Case II: E>VCase II: E>V00 (Scattering)(Scattering)L

Probability of reflection (Probability of reflection (reflection coefficientreflection coefficient) and ) and probability of transmission (probability of transmission (transmission coefficienttransmission coefficient))probability of transmission (probability of transmission (transmission coefficienttransmission coefficient))

Barrier Potential: E>VBarrier Potential: E>V00Barrier Potential: E>VBarrier Potential: E>V00Case II: E>VCase II: E>V00 (Scattering)(Scattering)This time transmission should not be a This time transmission should not be a surprise but there are other surprisessurprise but there are other surprisessurprise, but there are other surprises….surprise, but there are other surprises….

LL

As in Case I, “F” (in As in Case I, “F” (in ΨΨ33) is equal to 0 ) is equal to 0 because if the particle goes into regionbecause if the particle goes into regionbecause if the particle goes into region because if the particle goes into region (3) there is nothing to reflect it. It just (3) there is nothing to reflect it. It just keeps going.keeps going.

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

Boundary ConditionsBoundary ConditionsBoundary ConditionsBoundary Conditions

Transmission coefficientTransmission coefficientTransmission coefficientTransmission coefficient

If t t thi lt ith b k k i i d th tIf t t thi lt ith b k k i i d th tIf you want to compare this result with your book, keep in mind that:If you want to compare this result with your book, keep in mind that:

I R d In lecture:In Reed:

Resonance ScatteringResonance ScatteringResonance ScatteringResonance Scattering

T becomes 1 whenever kT becomes 1 whenever k22L=nL=nππ, because sin(n, because sin(nππ)=0.)=0.For E>VFor E>V00, T oscillates with energy!, T oscillates with energy!00, gy, gy(For E<V(For E<V00, T increases with energy, as expected) , T increases with energy, as expected)

At certain energies the barrier is transparent to the incident matterAt certain energies the barrier is transparent to the incident matter--wave. So wave. So there are certain energies for which T=1 exactly! This is called there are certain energies for which T=1 exactly! This is called resonance resonance scatteringscattering..

Why does T oscillate with E?Why does T oscillate with E?Why does T oscillate with E?Why does T oscillate with E?

Let’s look at the probability density for two extreme values of T:Let’s look at the probability density for two extreme values of T:p y yp y y

T oscillates because the solution in region (2) is oscillatory and T oscillates because the solution in region (2) is oscillatory and dependent on kdependent on k22, which depends on energy!, which depends on energy!

LL

pp 22, p gy, p gy

Operators: The HamiltonianOperators: The HamiltonianOperators: The HamiltonianOperators: The HamiltonianLet’s consider again the 1Let’s consider again the 1--dim., timedim., time--indep. S.E.:indep. S.E.:

ThTh H ilt ni n p r t rH ilt ni n p r t rThe The Hamiltonian operatorHamiltonian operator::

Operators and general form of Operators and general form of eigenvalue equation eigenvalue equation

An An operatoroperator does something to a function and returns a result.does something to a function and returns a result.Recall: Recall:

only certain function only certain function ΨΨ(x) will satisfy S.E. for a given V(x).(x) will satisfy S.E. for a given V(x).ΨΨ :: eigenfunctionseigenfunctions corresponding to V(x)corresponding to V(x)ΨΨnn : : eigenfunctionseigenfunctions corresponding to V(x)corresponding to V(x)EEn n : Energy : Energy eigenvalueseigenvalues corresponding corresponding to to ΨΨnn

So general form of So general form of eigenvalueeigenvalue equation can be thought of as:equation can be thought of as:(O ) ((O ) (Ei f iEi f i ) () (Ei lEi l ) () (Ei f iEi f i ))(Operator) ((Operator) (EigenfunctionEigenfunction) = () = (EigenvalueEigenvalue) () (EigenfunctionEigenfunction))

General example:General example:

Kinetic energy and potential Kinetic energy and potential energy operatorsenergy operators

We can consider the total energy operator HWe can consider the total energy operator HWe can consider the total energy operator HWe can consider the total energy operator Hopopto be the sum of the individual kinetic energy to be the sum of the individual kinetic energy (KE(KE ) and potential energy (PE) and potential energy (PE ) operators:) operators:(KE(KEopop) and potential energy (PE) and potential energy (PEopop) operators:) operators:

Eigenstate of energyEigenstate of energyEigenstate of energyEigenstate of energyIs Is the infinite the infinite square well square well solution an solution an eigenstateeigenstate of energy?of energy?

We will usethis wavefunctionthroughout today’slecture

Momentum operatorMomentum operatorMomentum operatorMomentum operatorLet’s determine the operator for linear momentum PLet’s determine the operator for linear momentum Popop

Recall a left or right propagating matterRecall a left or right propagating matter--wave:wave:Recall a left or right propagating matterRecall a left or right propagating matter wave: wave: (p>0: right propagating, p<0: left propagating)(p>0: right propagating, p<0: left propagating)

Reed is confused about the sign.

Eigenstate of momentumEigenstate of momentumEigenstate of momentumEigenstate of momentum

Is the infinite square well solution anIs the infinite square well solution an eigenstateeigenstateIs the infinite square well solution an Is the infinite square well solution an eigenstateeigenstateof momentum?of momentum?

So it is So it is notnot an an eigenstateeigenstate of momentum.of momentum.M i d f iM i d f i √(2 E√(2 E ) d i) d iMagnitude of momentum is Magnitude of momentum is ppnn=√(2mE=√(2mEnn) and is ) and is constant, but the direction is not determined; it constant, but the direction is not determined; it can be either left or rightcan be either left or rightcan be either left or right.can be either left or right.

Eigenstate of positionEigenstate of positionEigenstate of positionEigenstate of position

Is the infinite square Is the infinite square well solution an well solution an eigenstateeigenstate of of qq ggposition?position?

So it is So it is notnot an an eigenstateeigenstate of position.of position.

We find that the infinite square well waveWe find that the infinite square well wave--functions are functions are not not eigenstateseigenstates of momentum and position …. But we of momentum and position …. But we can evaluate something else whether or not it’s ancan evaluate something else whether or not it’s ancan evaluate something else, whether or not it s an can evaluate something else, whether or not it s an eigenstateeigenstate ……

Useful operatorsUseful operatorsUseful operatorsUseful operators

Reed, Chapter 4

Average valueAverage valueAverage valueAverage valueQuestion: what is the position of the particle?Question: what is the position of the particle?Answer: given in terms of a probability distributionAnswer: given in terms of a probability distributionA more meaningful question: what is the A more meaningful question: what is the average positionaverage position of the particle?of the particle?Example: what is the Example: what is the averageaverage score on an exam?score on an exam?

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

Reed, Chapter 4

Score in this example

ppii is the probability of a randomly chosen student to fall in bin iis the probability of a randomly chosen student to fall in bin iThe average of a quantity x is the sum (over all possible values of x) of x The average of a quantity x is the sum (over all possible values of x) of x g q y ( p )g q y ( p )times the probability of having x as the value times the probability of having x as the value

Expectation valueExpectation valueExpectation valueExpectation valueMore generally, if we consider position as a continuous variable More generally, if we consider position as a continuous variable and not a discrete one, we write the average value as the and not a discrete one, we write the average value as the

i li l ( l d d < >)( l d d < >)expectation value expectation value (also denoted as <x>):(also denoted as <x>):

Position expectation Position expectation value for value for infinite square wellinfinite square well

Position expectation Position expectation value value for for infinite square wellinfinite square well

This result means that average of many This result means that average of many measurements of the position would be at measurements of the position would be at x=L/2.x=L/2.//It is independent of n!It is independent of n!Well is symmetric, so particle does not prefer Well is symmetric, so particle does not prefer

id f ll h h hid f ll h h hone side of well to the other, no matter what one side of well to the other, no matter what state n it is in.state n it is in.Note that Note that ΨΨ is sometimes zero at x=L/2 !is sometimes zero at x=L/2 !So “expectation value” means “average value” So “expectation value” means “average value” not “most probable value”not “most probable value”

Momentum expectation Momentum expectation value value for for infinite square wellinfinite square well

See appendix C in Reed for useful integralsReed for useful integrals

Again is not surprising.Again is not surprising.The well is symmetric so the particle should have noThe well is symmetric so the particle should have noThe well is symmetric so the particle should have no The well is symmetric so the particle should have no preference for traveling one way or the other.preference for traveling one way or the other.

Expectation value of EnergyExpectation value of EnergyExpectation value of EnergyExpectation value of Energy

The expectation value of the energy for the infinite The expectation value of the energy for the infinite square well state n is just the eigenvalue of that state!square well state n is just the eigenvalue of that state!q j gq j g

Expectation value of pExpectation value of p22Expectation value of pExpectation value of p

The need for this will become clear later (nextThe need for this will become clear later (nextThe need for this will become clear later (next The need for this will become clear later (next time).time).

Revisit expectation value of Revisit expectation value of energyenergy

Revisit expectation value of Revisit expectation value of energyenergy

Expectation value of xExpectation value of x22Expectation value of xExpectation value of xAgain, this is something that we will find useful later.Again, this is something that we will find useful later.

Note: <xNote: <x22> is > is notnot equal to <x>equal to <x>22

Summary/AnnouncementsSummary/AnnouncementsSummary/AnnouncementsSummary/AnnouncementsIntroduced Operators and Expectation ValuesIntroduced Operators and Expectation ValuesNext time:Next time:Next time:Next time:

Uncertainty PrincipleUncertainty PrincipleCommutatorsCommutators

ΔxΔp ≥ h / 2

HW #4 due on Monday Oct 3!HW #4 due on Monday Oct 3!yy