Chapter06Quantum Mechanics II

General Bibliography

1) Various wikipedia, as specified2) Thornton-Rex, Modern Physics for Scientists & Eng, as

indicated

Postulates of Quantum Mechanics

• All information that can be known about a particle/system is contained within the wavefunction (x,t)

• The probability of finding the particle at (x,t) is given by.– in a region x about (x,t) byx

• The probability of finding the particle anywhere is 100% or 1.

spaceall

dx*1

Postulates of Quantum Mechanics• Every observable quantity is associated with an “Operator” which is

used to extract the information from the wavefunction

• The expectation value (aka average) of an observable quantity is given by

• The time evolution of the wavefunction is determined by

spaceall

dxOpOporOp *____

rrorrxxorx operatoroperator

ˆˆ

dt

diH

dx

diporp

dt

diEorE xoperatorxtotoperatortot ˆˆ

,,

H = Hamiltonian

Example Functions

• Can it be a solution to the time-dependent Schroedinger Eqn?

)sin( tkxA

)cos( tkxA

)( tkxieA

yetspecifiednotxwithex ti )()(

Properties of Good Wavefunctions

• must be finite everywhere

• must be single-valued

• and d/dx must be continuous for finite well-behaved potentials V(x)

• must be normalizable (generally 0 as xinf)

Infinite Square Well

Infinite Square Well

6.4 Finite Square Well

6.5 3-D Finite Square Well

6.6 Simple Harmonic Oscillator

For Etot less than barrier height