Physics 451 Quantum mechanics I Fall 2012 Oct 5, 2012 Karine Chesnel

Homework tonight! HW # 10 due Friday Oct 5 by 7pm 2.33, 2.34, 2.35 Announcements Quantum mechanics Homework next week: HW # 11 due Tuesday Oct 9 by 7pm 2.38, 2.39, 2.41, A1, A2, A5, A7 HW # 12 due Thursday Oct 11 by 7pm

Quantum mechanics Square wells and delta potentials V(x) x Bound states E < 0 Scattering States E > 0 x V(x) -V 0 -a +a V(x) x

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

Quantum mechanics Square wells and delta potentials Ch 2.6 Continuity at boundaries Delta functions Square well, steps, cliffs… is continuous is continuous except where V is infinite     m dx d         is continuous

Quantum mechanics Square wells and delta potentials Ch 2.6 Finding a solution Scattering states: Bound states Find the specific values of the energy Find the relationship between transmitted wave and incident wave Transmission coefficient Tunneling effect

Quantum mechanics Square barrier x V(x) V0V0 -a+a Ch 2.6 Pb. 2.33

Phys 451 The finite square barrier Scattering states V(x) x -V 0 A B F Coefficient of transmission for Pb. 2.33

Quantum mechanics “Step” potential and “cliff” Ch 2.6 V0V0 x V(x) -V0-V0 x Pb Pb Reflection coefficient R Different definition for the transmission coefficient T (use the probability current J) Analogy to physical potentials

Need for a formalism Quantum mechanics Wave function Vector Operator Matrix

Quantum mechanics Vectors i j k Physical spaceGeneralization (N-space) Addition - commutative - associative Scalar multiplication linear combination zero vector basis of vectors

Quantum mechanics Inner Product i j k Physical space  A B “Dot product” Generalization (N-space) “Inner product” Norm Orthogonality Orthonormal basis Schwarz inequality

Quantum mechanics Matrices Physical space i j k  A A’ Transformations: - Multiplication - rotation - symmetry… A’’ Generalization (N-space) Linear transformation Matrix Sum Product Unit matrix Inverse matrix Unitary matrix Transpose Conjugate Hermitian conjugate

Quantum mechanics Quiz 15 A matrix is “Hermitian” if: A. B. C. D. E.

Quantum mechanics Changing bases Physical space i j k i’ j’ k’ Generalization (N-space) Old basisNew basis Expressing same transformation T in different bases Same determinant Same trace

Quantum mechanics Formalism N-dimensional space: basis Operator acting on a wave vector:Expectation value/ Inner product Norm: For Hermitian operators: