Physics 48 – Quantum Mechanics

Larry Hunter120 Merrill Science CenterX- 2074LRHunter@amherst.eduLabs: 101, 104, 108 and 206 Merrill

Office Hours: Monday – 2 PM, Tuesday – 2 PM

Text: Introduction to Quantum Mechanics, Second Edition

David J. Griffiths, (Prentice Hall, 2005)Available at Amherst Books

What you know?

• Concept of complete orthonormal sets.• Some methods for solving partial

differential equations.• Quantum history – why it was necessary.

– Blackbody radiation – Plank and Leonard– Photo-electric effect – Einstein– Compton Effect– Pair Production

• Particle Wave Duality (see next slides).• Uncertainty Principle.• Bohr-Rutherford atomic model.• Schrödinger Equation in 1D

t dependent and independent.– Basic intuitions – what solutions should look

like and how they evolve. – Superposition.– Free particle solutions.– QM tunneling and barrier penetration.– Infinite square well.– Finite square well.

A Reminder about matter waves.

• In Physics 25 you discovered that light must at times be treated as a particle with E = hν and momentum p = E/c = hν/c = h/λ.

• DeBroiglie suggested that matter must have wave-like properties with

λ = h/p.This is seen in experiment.

Electron Scattering Image of a YIG Xtal

Molecular beam interference

Molecular Beam Interference Pattern

Spin Textures (Interference)

200 225 250 275 300 325 350 375 400

0.0

0.5

1.0

1.5

2.0

Opt

ical

Dep

th

position (μm)

•Anticorrelated•Fringe spacing ~ 25 μm•Confined to overlap region•Not observed if nointerconversion pulse applied

38o

A quasi-derivation of the S.E.• We demand that

and that

• We assume a non-relativistic limit such that for a free particle.

h//2 pk == λπ

./2 hE== πυω

./22 hmEkmEp =⇒=

We considered 4 possibilities and assume that the superposition principal must hold (this implies a linear equation – i.e. no terms like Ψ2).

( ) )sin(, kxttx −=Ψ ω

We look for plane wave solutions to describe a free particle

With only this solution the particle can not flicker in and out of existence.

For a Free particle moving in the +x direction

( ) ( )Ψ x t e i t kx, = − −ω

∂∂

∂∂

ΨΨ Ψ

ΨΨ

ti iE

E it

= − = −

⇒ =

ω / h

h

We can identify the energy operator t

iHδδ

= h

(also called the Hamiltonian, H)

The Momentum Operator

∂Ψ∂

∂Ψ∂

x ik ip

p ix

= =

⇒ = −

Ψ Ψ

Ψ

h

h

We identify the momentum operator as p ix

= − h∂∂

∂

∂2

22

2

22Ψ Ψ Ψ Ψ

xk

p mK= − = −

⎛⎝⎜

⎞⎠⎟ =

−h h

( ) ( )Ψ x t e i t kx, = − −ω

⇒ =

−Km x

ΨΨh2 2

22∂∂

We identify the kinetic energy operator Km x

=−h2 2

22∂∂

For a free particle we have Ψ=Ψ HK

With a potential we have the Schrodinger equation as a statement of conservation of energy: Ψ=Ψ+Ψ HUK

The Kinetic Energy Operator

Ta-Da!The time dependent Schrodinger Equation (TDSE)

( )( ) ( ) ( )

− + =h

h2 2

22mx t

xU x x t i

x tt

∂∂

∂Ψ∂

ΨΨ

,,

,

Note: The Schrödinger equation tells us how to handle conservative forces.Fortunately, nearly all microscopic forces are conservative! In this form we can not handle magnetic forces.

We assume that the potential is not time dependent.

The absolute value of the square of the wavefunction tells us about the probability of finding the particle at a position x at a time t.

( ) ( )P x t x t, ,= Ψ2

We require that the probability be normalized

( )P x t dx,

Normalization

−∞

∞∫ = 1

What about our free particle solution?

( ) ( )Ψ x t e i t kx, = − −ω

( ) ( ) ∞===Ψ= ∫ ∫∫∞

∞−

−− dxdxedxtxP kxti 22, ω ?????