Quantum mechanics I Fall 2012 Physics 451 Quantum mechanics I Fall 2012 Oct 4, 2012 Karine Chesnel

HW # 10 extended until Friday Oct 5 by 7pm Quantum mechanics Announcements Homework HW # 10 extended until Friday Oct 5 by 7pm Pb 2.33, 2.34, 2.35

The finite square well Quantum mechanics V(x) Scattering states -a a x Bound states -V0

The finite square well Quantum mechanics V(x) x -V0 Ch 2.6 Continuity at boundaries V(x) x -V0 is continuous X=+a X=-a

The finite square well Quantum mechanics Ch 2.6 Scattering state For Outside the well For For

The finite square well Quantum mechanics Ch 2.6 Scattering state For General solution

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 -V0 Continuity at boundaries V(x) x C,D A F B -V0 at x = +a at x = -a Continuity of Continuity of

The finite square well Quantum mechanics x -V0 Finally Continuity at boundaries V(x) x C,D A F B -V0 Finally

The finite square well Quantum mechanics x -V0 A F B Scattering state V(x) x A F B -V0 Coefficient of transmission

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

Quiz 14 True B. False Quantum mechanics We have seen that the coefficient of transmission oscillates with energy, and that the well becomes ‘transparent” for a particle in a scattering state when its energy equals specific values En. Similarly, we can show that the coefficient of reflection oscillates and the well becomes like a perfect wall, so the particle is totally reflected for some other specific values of energy En’. True B. False

Transmission versus energy Ch 2.5 Quantum mechanics Transmission versus energy Transmission coefficient Delta function well

Square wells and delta potentials Quantum mechanics Square wells and delta potentials V(x) x Scattering States E > 0 V(x) x x V(x) -V0 -a +a Bound states E < 0

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 Ch 2.6 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

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

Ch 2.6 Quantum mechanics Square barrier V(x) V0 x -a +a Pb. 2.33

The finite square barrier Phys 451 The finite square barrier Scattering states V(x) x -V0 A B F Pb. 2.33 for Coefficient of transmission for for

transmission coefficient Ch 2.6 Quantum mechanics “Step” potential and “cliff” V0 x V(x) -V0 x V(x) Pb. 2.35 Pb. 2.34 With a different definition for the transmission coefficient Analogy to physical potentials