Quantum mechanics I Fall 2012 Physics 451 Quantum mechanics I Fall 2012 Review 2 Karine Chesnel

EXAM II When: Tu Oct 23 – Fri Oct 26 Where: testing center Quantum mechanics EXAM II When: Tu Oct 23 – Fri Oct 26 Where: testing center Time limited: 3 hours Closed book Closed notes Useful formulae provided

EXAM II 1. The delta function potential 2. The finite square potential Quantum mechanics EXAM II 1. The delta function potential 2. The finite square potential (Transmission, Reflection) 3. Hermitian operator, bras and kets 4. Eigenvalues and eigenvectors 5. Uncertainty principle

Square wells and delta potentials Quantum mechanics Square wells and delta potentials V(x) Physical considerations Scattering States E > 0 x Symmetry considerations Bound states E < 0

( ) Square wells and delta potentials 2 y a h m dx d - = ÷ ø ö ç è æ D Quantum mechanics Square wells and delta potentials Continuity at boundaries is continuous is continuous except where V is infinite ( ) 2 y a h m dx d - = ÷ ø ö ç è æ D Delta functions Square well, steps, cliffs… is continuous

The delta function potential Quantum mechanics The delta function potential For

The delta function well Quantum mechanics The delta function well Bound state

The delta function well/ barrier Ch 2.5 Quantum mechanics The delta function well/ barrier Scattering state A F B x “Tunneling” Reflection coefficient Transmission coefficient

The finite square well Quantum mechanics x -V0 Bound state Symmetry considerations The potential is even function about x=0 V(x) The solutions are either even or odd! x -V0

Quantum mechanics The finite square well Bound states where

The finite square well Quantum mechanics x -V0 Scattering state +a -a (2) (1) V(x) (3) -a +a x A B F C,D (1) (3) (2) -V0

The finite square well Quantum mechanics x The well becomes transparent (T=1) when V(x) x A F B -V0 Coefficient of transmission

Linear transformation Quantum mechanics Formalism Wave function Vector Linear transformation (matrix) Operators Observables are Hermitian operators

Eigenvectors & eigenvalues Quantum mechanics Eigenvectors & eigenvalues For a given transformation T, there are “special” vectors for which: is an eigenvector of T l is an eigenvalue of T

Eigenvectors & eigenvalues Quantum mechanics Eigenvectors & eigenvalues To find the eigenvalues: We get a Nth polynomial in l: characteristic equation Find the N roots Spectrum Find the eigenvectors

Hilbert space Quantum mechanics Infinite- dimensional space N-dimensional space Hilbert space: functions f(x) such as Inner product Orthonormality Schwarz inequality

The uncertainty principle Quantum mechanics The uncertainty principle Finding a relationship between standard deviations for a pair of observables Uncertainty applies only for incompatible observables Position - momentum

The uncertainty principle Quantum mechanics The uncertainty principle Energy - time Derived from the Heisenberg’s equation of motion Special meaning of Dt

Different notations to express the wave function: Quantum mechanics The Dirac notation Different notations to express the wave function: Projection on the energy eigenstates Projection on the position eigenstates Projection on the momentum eigenstates

The Dirac notation Bras, kets Operators, projectors Quantum mechanics = inner product = matrix (operator) Operators, projectors projector on vector en