Quantum Physics IIPHYS 402, Fall 2015 Instructor: Victor Galitski

Lecture 2: A physical interpretation of QMPart I: Meaning of the wave function; the Born rule

Schrödinger equation

The Nobel Prize in Physics 1933 was awarded jointly to Erwin Schrödinger and Paul Adrien Maurice Dirac "for the discovery of new productive forms of atomic theory."

Born interpretation

The Nobel Prize in Physics 1954 was divided equally between Max Born "for his fundamental research in quantum mechanics, especially for his statistical interpretation of the wavefunction"

So, what is the wave-function?

• It may be a wrong question to ask, as the wave-function is a mathematical construct(one among several others) that allows us to calculate what we are interested in (observables, i.e., something we can actually measure).

Quantum observable, X, at t=0

Know:

Quantum observable, X, at a later time t=T

Want to know:

No closed equation for X…

Find the wave-functionwhich satisfies a closed

Eq and determines X

• We certainly have seen such auxiliary concepts elsewhere in physics. For example:

Neither scalar nor vector potential is directly observable, which does not surpriseus, neither should we be surprised by the wave-function…

Quantum Physics IIPHYS 402, Fall 2015 Instructor: Victor Galitski

Lecture 2: A physical interpretation of QMPart II: Operators

Appearance of operators in Schrödinger’s “derivation”

Experimental fact: quantum electronsmay exhibit wave-like properties

Assume that they may be describedby a plane-wave function:

We have to reconcile it with

Hence the free Schrodinger equationWe can write it as

if we identify,

Generalize it to

with

Expectation values

Quantum Physics IIPHYS 402, Fall 2015 Instructor: Victor Galitski

A physical interpretation of quantum theoryPart III: Time-independent Schrödinger Eq. Eigenvalue problems

Getting rid of the time derivative, when it’s not needed

A simple (mathematical) example of an eigenvalue problem

Operators, eigenvalues, and eigenvectors in QM: summary

Quantum Physics IIPHYS 402, Fall 2015 Instructor: Victor Galitski

Lecture 2: A physical interpretation of QMPart IV: Superposition principle; Dirac notations; representations

Superposition principle in quantum mechanics

Simple reminder from linear algebra

x

y

x’

y’

How to choose a basis/representation