Quantum Mechanics 1D

Bright point that appears at the centre of a circular objects shadow

Arago spot

Ultra violet catastrophe

Rayleigh- jeans law fails at low wavelengths

E=

(Frequency, angular frequency, wavelength, momentum)

E=hf

E=hbar omega

E=hc/lambda

P=E/c

De broglie wavelength

Lambda=h/p

Scale where quantum effects are significant

Wavefunction

Psi(x,t)= Aexp[i/h (px-Et)]

Heisenberg’s uncertainty principle

Deltaxdeltap >=hbar/2

P(a,b)=

Integral of psi ²

Bra-ket

Integral of the bra x ket

State postulate

At any point in time, the state of a quantum mechanical system is completely specified by a function psi.

Observable

Measurable quantity

Expectation value

Average value obtained when an observable is repeatedly measured. Must be real

Expectation value for position

<psi|x|psi>

Momentum expectation value

<psi|P|psi> where P is -i hbar d/dx

Operator postulate

Any observable A has an associated linear operator A hat

Expectation postulate

The expectation value of an observable A is given by <psi| A hat|psi>

Uncertainty in a measurement

Sqrt[ <A²> - <A>² ]

Total energy

Hamiltonian = Ť + V = -(hbar²/2m)d²/dx² + V(x,t)

Evolution postulate

The wavefunction of a system evolves in time according to the time-dependent Schrödinger equation HPsi = iħ dPsi/dt