Modern Physics (II) Chapter 9: Atomic Structure Chapter 10: Statistical Physics Chapter 11: Molecular Structure Chapter 12-1: The Solid State Chapter 12-2: Superconductivity Serway, Moses, Moyer: Modern Physics Tipler, Llewellyn: Modern Physics

Modern Physics I Chap 3: The Quantum Theory of Light Blackbody radiation, photoelectric effect, Compton effect Chap 4: The Particle Nature of Matter Rutherford’s model of the nucleus, the Bohr atom Chap 5: Matter Waves de Broglie’s matter waves, Heisenberg uncertainty principle Chap 6: Quantum Mechanics in One Dimension The Born interpretation, the Schrodinger equation, potential wells Chap 7: Tunneling Phenomena (potential barriers) Chap 8: Quantum Mechanics in Three Dimensions Hydrogen atoms, quantization of angular momentums

Chapter 5: Matter Waves de Broglie’s intriguing idea of “matter wave” (1924) Extend notation of “wave-particle duality” from light to matter For photons, The wavelength is detectable only for microscopic objects Suggests for matter, de Broglie wavelength P: relativistic momentum de Broglie frequency E: total relativistic energy

Chapter VI Quantum Mechanics in One Dimension (x,t ) contains within it all the information that can be known about the particle (x, t ) is an infinite set of numbers corresponding to the wavefunction value at every point x at time t Properties of wavefunction Finite, single-valued, and continuous on x and t The particle can be found somewhere with certainty Normalization: must be “smooth” and continuous where U(x) has a finite value

Time-independent Schrödinger equation The one-dimensional Schrödinger wave equation Time-independent Schrödinger equation U(x,t ) = U(x), independent of time E: total energy of the particle Solution: Probability density at any given position x (independent of time) stationary states

Time-independent Schrödinger equation Separation of variables : (x,t ) = (x)·(t ) = E = constant Independent of t Independent of x spatial temporal

A particle in a finite square well U(x) L U0 I II III Region II Region I, III x Need to solve Schrodinger wave equation in regions I, II, and III Consider: E < U0 The wavefunctions look very similar to those for the infinite square well, except the particle has a finite probability of “leaking out” of the well  > 0 Region I: Region II: Region III:

Finite square Well Penetration depth n = 2 n = 3 U(x) L U0 I II III x L U0 I II III x n=1 y(x) U0 n = 3 U(x) I II III x Penetration depth No classical analogy !!

Example: A particle in an infinite square well of width L Momentum is quantized. Energy is quantized ! The notion of quantum number: n

Chapter VII Tunneling Phenomena The Square Barrier Potential U(x) x Uo L I II III Resonance transmission at certain energies E > U0 A finite transmission through the barrier at E < U0 if the barrier is made sufficiently thin

Expectation values Observables (可觀測量) and Operators (算符) For a given wavefunction (x,t ), there are two types of measurable quantities: eigenvalues, expectation values Observables (可觀測量) and Operators (算符) An “observable” is any particle property that can be measured Expectation value Q predicts the average value for Q  (x,t ) is the “eigenfunction” and q is the “eigenvalue” q = constant The Schrödinger wave equation:

Examples of eigenvalues and eigenfunctions U = central forces U = 0, a free particle

Chapter VIII Quantum Mechanics in Three Dimensons Three-dimensional Schrödinger equation Time-independent Schrödinger equation:

Particle in a system with central forces electron  r nucleus a central force !!  Require use of spherical coordinates

ml l n Time-independent Schrödinger equation principal quantum number orbital quantum number magnetic quantum number ml l n

Since |L| and Lz are quantized differently, L cannot orient in the For any central force U(r ), angular momentum is quantized by the rules and Since |L| and Lz are quantized differently, L cannot orient in the z-axis direction. |L| > Lz  = 1, 2, 3, … (n-1) ml = 0, 1, 2, …  Degeneracy for a given n n = 1, 2, 3,…

Probability of finding electron of a hydrogen-like atom in the spherical shell between r and r + dr from the nucleus 0.52 Å l = 0

Excited States of Hydrogen-like Atoms The first excited state: n = 2 fourfold degenerate 2s state, is spherically symmetric 2p states, is not spherically symmetric

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