PHYS274 Quantum Mechanics VII Further basics of Quantum Mechanics Tunneling, Harmonic Oscillator, Complete Chap 40

Potential barriers and tunneling The figure (below left) shows a potential barrier. In Newtonian physics, a particle whose energy E is less than the barrier height U0 cannot pass from the left-hand side of the barrier to the right-hand side. Figure (below right) shows the wave function Ψ (x) for such a particle. The wave function is nonzero to the right of the barrier, so it is possible for the particle to “tunnel” from the left-hand side to the right-hand side.

Applications of tunneling A scanning tunneling microscope measures the atomic topography of a surface. It does this by measuring the current of electrons tunneling between the surface and a probe with a sharp tip (see the Fig. below). An alpha particle inside an unstable nucleus can only escape via tunneling (will study a bit more in Nuclear Physics chapter) Also tunnel diodes, Josephson junctions (superconducting)

Biophysics: Tunneling is very important in enzymes Cartoon illustration of methylamine dehydrogenase (MADH) suspended in water, showing the active site of the enzyme in the midst of the protein as spheres. Tunneling of electrons from one part of the molecule to the other. Without quantum mechanical tunneling, “life as we know it would be impossible” p.1349 of Freedman and Young

Example of a tunneling calculation Tunneling probability A 2.0 eV electron encounters a rectangle barrier 5.0 eV high. What is the tunneling probability if the barrier width is 0.50 nm ? Find G and κ 5

Example of tunneling calculation Tunneling probability A 2.0 eV electron encounters a rectangle barrier 5.0 eV high. What is the tunneling probability if the barrier width is 0.50 nm ? 6

Example of tunneling calculation Tunneling probability A 2.0 eV electron encounters a rectangle barrier 5.0 eV high. What is the tunneling probability if the barrier width is 0.50 nm ? Reducing the width L by a factor of two, increases the tunnel probability of a factor of 7300. Exponential sensitivity in the barrier factor. 7

Clicker Tunneling Question A potential-energy function is shown. If a quantum-mechanical particle has energy E < U0, it is impossible to find the particle in the region A. x < 0. B. 0 < x < L. C. x > L. D. misleading question—the particle can be found at any x Answer: D 8

Clicker Tunneling Question A potential-energy function is shown. If a quantum-mechanical particle has energy E < U0, it is impossible to find the particle in the region A. x < 0. B. 0 < x < L. C. x > L. D. misleading question—the particle can be found at any x Answer: D 9

A comparison of Newtonian and quantum oscillators Classical simple harmonic oscillator. Let’s work out Ψ (x) for the QM harmonic oscillator. 10

A comparison of Newtonian and quantum oscillators Let’s write the Schrodinger equation for the QM harmonic oscillator. 11

A comparison of Newtonian and quantum oscillators Let’s work out Ψ (x) for the QM harmonic oscillator. Why is (d) the only possibility ? Other violate unitarity, or have negative probability (a), (b), and c all go to infinity or –infinity as xinfinity 12

A comparison of Newtonian and quantum oscillators Using the boundary condition, we obtain Ψ (x) for the QM harmonic oscillator and its energy levels. These are the possible energies of the QM harmonic oscillator Note that ω=√(k’/m) but that n starts from n=0 !! (a), (b), and c all go to infinity or –infinity as xinfinity The ground state has n=0; no QM solution with energy equal to zero !! 13

A comparison of Newtonian and quantum oscillators Figure (below, top) shows the first four stationary-state wave functions Ψ (x) for the harmonic oscillator. A is the amplitude of oscillation in Newtonian physics. Figure (below, bottom) shows the corresponding probability distribution functions |Ψ (x)|2. The blue curves are the Newtonian probability distributions.

A comparison of Newtonian and quantum oscillators Classical, confined to –A, A; The QM SHM is different (not confined). Question: What is meant by not –confined ? 15

Clicker question on QM Harmonic Oscillator The figure shows the first six energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics. B. do not have a definite wavelength. C. are all equal to zero at x = 0. D. Both A. and B. are true. E. All of A., B., and C. are true. Answer: D 16

A. are nonzero outside the region allowed by Newtonian mechanics. QM SHO The figure shows the first six energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics. B. do not have a definite wavelength. C. are all equal to zero at x = 0. D. Both A. and B. are true. E. All of A., B., and C. are true. 17

an infinitely deep square potential well (particle in a box) Q27.1 QM Potential Wells A particle in a potential well emits a photon when it drops from the n = 3 energy level to the n = 2 energy level. The particle then emits a second photon when it drops from the n = 2 energy level to the n = 1 energy level. The first photon has the same energy as the second photon. What kind of potential well could this be? an infinitely deep square potential well (particle in a box) B. a harmonic oscillator C. either A. or B. D. neither A. nor B. Answer: B 18

A. an infinitely deep square potential well (particle in a box) Q27.1 QM Potential Wells A particle in a potential well emits a photon when it drops from the n = 3 energy level to the n = 2 energy level. The particle then emits a second photon when it drops from the n = 2 energy level to the n = 1 energy level. The first photon has the same energy as the second photon. What kind of potential well could this be? A. an infinitely deep square potential well (particle in a box) B. a harmonic oscillator C. either A. or B. D. neither A. nor B. 19

Q27.2 B 20

Q27.2 B 21

Q27.3 C 22

Q27.3 C 23

Q27.4 B 24

Q27.4 B 25

Q27.5 C 26

Q27.5 C 27

Read about the Schrodinger Equation in 3 dimensions and 3-D particle in a box for next time. 28