Quantum Mechanics I Erwin Schrodinger Richard Feynman bongo

Quantum Mechanics I Erwin Schrodinger Richard Feynman, bongo drummer and physicist • Quiz – Full-on Quantum Mechanics part I 1

Q 21. 1 In order for a proton to have the same momentum as an electron, A. the proton must have a shorter de Broglie wavelength than the electron B. the proton must have a longer de Broglie wavelength than the electron C. the proton must have the same de Broglie wavelength as the electron D. not enough information given to decide 2

Q 21. 1 In order for a proton to have the same momentum as an electron, A. the proton must have a shorter de Broglie wavelength than the electron B. the proton must have a longer de Broglie wavelength than the electron C. the proton must have the same de Broglie wavelength as the electron D. not enough information given to decide λ = h/p 3

Q 21. 2 Which of the following statements is correct? A. Spontaneous emission emits photons in random directions at random times B. Both spontaneous emission and stimulated emission emit photons in random directions and random times C. Both spontaneous emission and stimulated emission emit photons in the same direction 4

Q 21. 2 Which of the following statements is correct? A. Spontaneous emission emits photons in random directions at random times B. Both spontaneous emission and stimulated emission emit photons in random directions and random times C. Both spontaneous emission and stimulated emission emit photons in the same direction 5

Q 21. 3 As you increase the temperature of a blackbody, which of the following statements is true? A. B. C. D. It radiates more high frequency photons It radiates fewer high frequency photons Its radiation spectrum is independent of temperature I(λ)~λ 4 6

Q 21. 3 As you increase the temperature of a blackbody, which of the following statements is true? A. B. C. D. It radiates more high frequency photons It radiates fewer high frequency photons Its radiation spectrum is independent of temperature I(λ)~λ 4 7

Review: Continuous spectra and blackbody radiation-Classical • A blackbody is an idealized case of a hot, dense object. The continuous spectrum produced by a blackbody at different temperatures is shown on the right. A classical Physics calculation by Lord Rayleigh gives, Agrees quite well at large values of the wavelength λ but breaks down at small values. This is the “ultraviolet catastrophe” of classical physics. Question: why the colorful name ? 8

Review: Continuous spectra and blackbody radiation-QM A calculation by Max Planck assuming that each mode in the blackbody has E = hf gives, (he says this “was an act of desperation”) Agrees quite well at all values of wavelength and avoids the “ultraviolet catastrophe” of classical physics. At large wavelengths, Planck’s result agrees with Rayleigh’s formula. 9

“I have one simple request, and that is to have sharks with frickin' laser beams attached to their heads!” – Dr. Evil Compare difference light sources. What do they have in common? How are they different? 10

sources of light (traditional): light bulb filament Hot electrons. very large # close energy levels (metal) Radiate spectrum of colors. Mostly IR. atom discharge lamps Electron jumps to lower levels. Only specific wavelengths. 120 V or more with long tube IR P λ laser fluorescent 11

The laser • Atoms spontaneously emit photons of frequency f when they transition from an excited energy level to a lower level. • Excited atoms can be stimulated to emit coherently if they are illuminated with light of the same frequency f. This happens in a laser (Light Amplification by Stimulated Emission of Radiation) discovered by A. Einstein in 1916 (one γ in two γ’s out) 12

The laser Need a “population inversion” in state E 2, which is relatively long-lived (10 -3 s) compared to E 3 or E 1 (10 -8 s). Stimulated emission from E 2 to E 1 Note that the mirrors and resonant cavity are needed. 13

Getting a population inversion need at least one more energy level involved. Trick: use a second color of light 2 also can kick up by bashing with electron To create population inversion between G and level 1 would need: t 2 t 1 1 a. time spent in level 2 (t 2) before spontaneously jumping to 1 is long, and time spent in level 1 (t 1) before jumping to G is short. b. t 1=t 2 c. t 2 short, t 1 long G d. does not matter “pumping” process to produce population inversion 14

Getting a population inversion need at least one more energy level involved. Trick: use a second color of light 2 also can kick up by bashing with electron To create population inversion between G and level 1 would need: t 2 t 1 1 a. time spent in level 2 (t 2) before spontaneously jumping to 1 is long, and time spent in level 1 (t 1) before jumping to G is short. b. t 1=t 2 c. t 2 short, t 1 long G d. does not matter “pumping” process to produce population inversion 15

The Heisenberg Uncertainty Principle revisited • The Heisenberg uncertainty principle for momentum and position applies to electrons and other matter, as does the uncertainty principle for energy and time. This gives insight into two-slit interference with electrons A common misconception is that the interference pattern is due to two electrons passing through the slits at the same time and their waves interfere. We observe the same interference by passing one electron through the slits at a time and collect the data over time. 16

The Uncertainty Principle and the Bohr model An electron is confined within a region of width 5. 0 x 10 -11 m (the Bohr radius) a) Estimate the minimum uncertainty in the x-component of the electron’s momentum b) What is the kinetic energy of an electron with this Kinetic energy magnitude of momentum ? comparable to atomic energy levels ! 17

The Heisenberg Uncertainty Principle: • The Heisenberg uncertainty principle for momentum and position applies to electrons and other matter, as does the uncertainty principle for energy and time. 18

The Uncertainty Principle and spectral line width A sodium atom in an excited state remains in that state for 1. 6 x 10 -8 s before making a transition to the ground state. What is the uncertainty in energy for the excited state ? A. B. C. D. E. 0. 21 e. V 6. 41 x 10 -5 e. V 2. 1 x 10 -8 e. V 1. 6 x 10 -19 e. V 2. 41 x 10 -55 e. V 19

The Uncertainty Principle and spectral line width A sodium atom in an excited state remains in that state for 1. 6 x 10 -8 s before making a transition to the ground state. What is the uncertainty in energy for the excited state ? A. B. C. D. E. 0. 21 e. V 6. 41 x 10 -5 e. V 2. 1 x 10 -8 e. V 1. 6 x 10 -19 e. V 2. 41 x 10 -55 e. V 20

Answer: The Uncertainty Principle and spectral line width A sodium atom in an excited state remains in that state for 1. 6 x 10 -8 s before making a transition to the ground state. What is the uncertainty in energy for the excited state ? 21

The Heisenberg Uncertainty Principle revisited A sodium atom in an excited state remains in that state for 1. 6 x 10 -8 s before making a transition to the ground state. The photons emitted from a collection of sodium atoms have an average wavelength of 589. 0 nm and average energy of 2. 105 e. V Question: What is the fractional wavelength spread of the corresponding spectral line ? 22

QM Chapter 40: Wave functions in classical physics • This is the wave equation for “waves on a string” where y(x, t) is the displacement of the string at location x. • Notice this is a partial differential equation with v = velocity of wave propagation. There are similar wave equations for the E and B fields in E+M waves. End of class, will continue at next class 23

The Schrödinger equation in 1 -D • In a one-dimensional model, a quantum-mechanical particle is described by a wave function Ψ(x, t). [QM: remember point particles are waves] • The one-dimensional Schrödinger equation for a free particle of mass m is • The presence of i (the square root of – 1) in the Schrödinger equation means that wave functions are always complex functions. • The square of the absolute value of the wave function, |Ψ(x, t)|2, is called the probability distribution function. |Ψ(x, t)|2 dx tells us about the probability of finding the particle somewhere between location x and x+dx at time t. How do you calculate the total probability of finding the particle anywhere? • Warning: |Ψ(x, t)|2 is not a probability, |Ψ(x, t)|2 dx is. 24

The Schrödinger equation in 1 -D: A free particle • If a free particle has definite momentum p and definite energy E, its wave function (see the Figure) is • Calculate |Ψ(x, t)|2. • Is such a particle localized? Question: What is k ? Ans: It is the wavenumber, remember k=2π/λ How is the particle’s momentum and energy related to k and w? 25

The Schrödinger equation in 1 -D: A free particle If a free particle has definite momentum p and definite energy E, its wave function is N. B. this is non-relativistic 26

For next time • Getting to heart of Quantum Mechanics Ø Read material in advance Ø Concepts require wrestling with material 27