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