Quantum Mechanics And Atomic Physics

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Quantum Mechanics andAtomic PhysicsLecture 10:Orthogonality,, Superposition, TimeOrthogonalityTime-dependentwave functionsfunctions, f. Sean Oh

Last time: The UncertaintyPrinciple Revisited Heisenberg’s Uncertainty principle:ΔxΔp h / 2 Position and momentum do not commute If we measure the particle’s position more and moreprecisely, that comes with the expense of the particle’smomentum becomingbi lless andd lless wellll kknown.And vicevice--versa.

Ehrenfest’ss TheoremEhrenfest The expectation value of quantum mechanics followsthe equation of motion of classical mechanics.In classical mechanics In quantum mechanics, See Reed 4.5 for the proof.Average of many particles behaves like a classicalparticle

Orthogonality Theorem: EigenfunctionsTheorem:g fwith differentffeigenvaluesgare orthogonal.gConsider a set of wavefunctions satisfying the time independentS.E. for some potential V(x)Then orthogonality states:In other words, if any two members of the set obey the aboveintegral constraint, they constitute an orthogonal set ofwavefunctions.wavefunctionsLet’s prove this

Proof: Orthogonality Theorem

Proof, concon’tt

Theorem is proven

Orthonormality In addition, if each individual member of the set ofwavefunctions is normalized,, theyy constitute an orthonormalset:Kronecker delta

Degenerate Eigenfunctions If n kk, but En Ek, then we say that theeigenfunctions are degenerateSince En-Ek 0,0 the integralneed not be zeroBut it turns out that we can always obtainanother set of Ψ’s, linear combinations of theoriginals, such that the new Ψ’s are orthogonal.

Principle of Superposition Any linear combination of solutions to the timetime-dependent S.E. is also asolution of the T.D.S.E.F example,Forl particleti l ini infinitei fi it square wellll can beb ini a superpositioniti offstates:We covered this in lecture 4! T.D.S.E is:

Principle of Superposition II Is any linear combination of solutions to the timetime-independent S.E. also asolution of the T.I.S.E? In other words, linear combinations of eigenstates are not generally solutions ofthe eigenequationeigenequation.The measurement will yield either E1 or E2, though not with equal probabilityThe system need not be in an eigenstate - the superposition state Ψ “collapses” into onegwhen one makes a measurement to determine which state theof the eigenstatessystem is actually in.

Principle of Superposition III Ifare energy eigenfunctionseigenfunctions,, that is thesolution of the T.I.S.E. and the wavefunction attt 00 is given by, then ata later time t, the wavefunction is given by, whereEi is the eigenenergy corresponding toThen,, the expectationpvalue of the energygy isgiven by

Principle of Superposition IV Normalization also requires that

A TimeTime-Dependent WaveWave-Packet See Reed Section 4.8 for a very niceexample:Illustrates concept of a travelingwave and the principle of superposition

Measurement and wavefunctioncollapse Consider the infinite potential well problem.problemIf at t 0 Then at a later time t, At t 0, if you measure the energy of the system,what energy values can you measure with whatprobabilities?

Continued Now, if your measurement yielded E1, what isthe new wavefunction afterwards? Now, after this measurement, if you measure theenergy again,again what are the possible energy valueswith what probabilities?

Measurement continued Now if you measure the position of the particle,particlewhat position would you measure with whatprobabilities? Now if your measurement yielded x 4/L, andthen if you measure energy again, what energyvalues are possible?

Theorem If Ψ is in an eigenstate of Qop with eigenvalue λ,then Q λ and ΔQ 0.So, λ is the only value we’ll observe for Q!Proof: No uncertainty! Observe λ only.

Virial Theorem The Virial Theorem (VT) is an expression that relates the expectation valuesof the KEop and PEop for any potential.Suppose operator A is timetime-independent In VT, A is defined as: Section 4.9 in Reed goes through the proof of the VT in great detail whichgives:

Example: VT using a radialpotential

VT for Coulomb PotentialConsistent with the Bohr model

Summary/Announcements We covered various things today: OrthogonalityOrthogonality,,Superposition, Measurement, TimeTime-dependentwave function,function and various theoemsTime for quiz: Closed book, and closed note !Midterm exam Wed. Oct. 19 in class - it will be closedbook with a letter size formulaformula-ONLY ((no solutions,, orextra texts allowed) sheetsheet- Need to turn in togetherwith the answer book.

EhrenfestEhrenfest s’s Theorem The expectation value of quantum mechanics followsThe expectation value of quantum mechanics follows the equation of motion of classical mechanics. In classical mechanics In quantum mechanics, See Reed 4.5 for the proof. Av

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