QM Review and SHM in QM Review and

QM Review and SHM in QM Review and Tunneling Calculation. No quiz but some clicker questions. Read about the Schrodinger Equation in 3 dimensions and 3 -D particle in a box for next time. Copyright © 2012 Pearson Education Inc.

Review: The Schrödinger equation in 1 -D: Stationary states • If a particle of mass m moves in the presence of a potential energy function U(x), the one-dimensional Schrödinger equation for the particle is • If the particle has a definite energy E, the wave function Ψ(x, t) is a product of a time-independent wave function Ψ (x) and a factor that depends on time t but not position. For such a stationary state the probability distribution function |Ψ (x, t)|2 = |Ψ (x)|2 does not depend on time. • The time-independent one-dimensional Schrödinger equation for a stationary state of energy E is Copyright © 2012 Pearson Education Inc.

Review: Particle in a finite potential well I • A finite potential well is a region where potential energy U(x) is lower than outside the well, but U(x) is not infinite outside the well (see Figure below). • In Newtonian physics, a particle whose energy E is less than the “height” of the well can never escape the well. In quantum mechanics the wave function of such a particle extends beyond the well, so it is possible to find the particle outside the well. Copyright © 2012 Pearson Education Inc.

Particle in a finite potential well II • Figure (bottom left) shows the stationary-state wave functions Ψ (x) and corresponding energies for one particular finite well. • Figure (bottom right) shows the corresponding probability distribution functions |Ψ (x)|2. Copyright © 2012 Pearson Education Inc.

Application of QM potentials: QM semiconductor dots Nanometer-sized (nm) particles of a semiconductor such as Cd. Se Discovered by Russian physicist Alexey Ekimov in St. Petersburg in 1981 The fluorescence color under UV is determined by the size of the dot. (5 -6 nm orange or red; 2 -3 nm blue or green) Can be injected into patients for medical research and treatment (tagging and treating cancer cells). Also solar cell and computing applications Copyright © 2012 Pearson Education Inc.

QM Potential Well Clicker The first three wave functions for a finite square well are shown. The probability of finding the particle at x > L is A. least for n = 1. B. least for n = 2. C. least for n = 3. D. the same (and nonzero) for n = 1, 2, and 3. E. zero for n = 1, 2, and 3. Copyright © 2012 Pearson Education Inc.

QM Potential Well Clicker The first three wave functions for a finite square well are shown. The probability of finding the particle at x > L is A. least for n = 1. B. least for n = 2. C. least for n = 3. D. the same (and nonzero) for n = 1, 2, and 3. E. zero for n = 1, 2, and 3. Copyright © 2012 Pearson Education Inc.

Review: 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 U 0 cannot pass from the left-hand side of the barrier to the right-hand side. • Look at 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. Copyright © 2012 Pearson Education Inc.

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 Figure below). • An alpha particle inside an unstable nucleus can only escape via tunneling (see Figure on the right). UH Physics Professor Klaus Sattler has a scanning tunneling microscope Copyright © 2012 Pearson Education Inc.

Example of tunneling calculation Tunneling probability A 2. 0 e. V electron encounters a rectangle barrier 5. 0 e. V high. What is the tunneling probability if the barrier width is 0. 50 nm ? Find G and κ Copyright © 2012 Pearson Education Inc.

Example of tunneling calculation Tunneling probability A 2. 0 e. V electron encounters a rectangle barrier 5. 0 e. V high. What is the tunneling probability if the barrier width is 0. 50 nm ? Copyright © 2012 Pearson Education Inc.

Example of tunneling calculation Tunneling probability A 2. 0 e. V electron encounters a rectangle barrier 5. 0 e. V high. What is the tunneling probability if the barrier width is 0. 50 nm ? Reducing the width L by a factor of two, increasing the tunnel probability of a factor of 7300. Exponential sensitivity in the barrier factor. Copyright © 2012 Pearson Education Inc.

Clicker Tunneling Question A potential-energy function is shown. If a quantum-mechanical particle has energy E < U 0, 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 Copyright © 2012 Pearson Education Inc.

Clicker Tunneling Question A potential-energy function is shown. If a quantum-mechanical particle has energy E < U 0, 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 Copyright © 2012 Pearson Education Inc.

A comparison of Newtonian and quantum oscillators • Let’s work out Ψ (x) for the QM harmonic oscillator. Copyright © 2012 Pearson Education Inc.

A comparison of Newtonian and quantum oscillators • Let’s work out Ψ (x) for the QM harmonic oscillator. Why is (d) the only possibility ? Copyright © 2012 Pearson Education Inc.

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 ! The ground state has n=0; no QM solution with energy equal to zero. Copyright © 2012 Pearson Education Inc.

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. Copyright © 2012 Pearson Education Inc.

A comparison of Newtonian and quantum oscillators Classical, confined to –A, A; QM SHM is different. � Copyright © 2012 Pearson Education Inc.

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. Copyright © 2012 Pearson Education Inc.

A 40. 8 The figure shows the first six energy levels of a quantummechanical 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. Copyright © 2012 Pearson Education Inc.

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. Copyright © 2012 Pearson Education Inc.

A 40. 9 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. Copyright © 2012 Pearson Education Inc.