1Chapter Objectives Mathematics Concepts How is a coordinate system chosen? What is the meaning of the divergence of a vector quantity? How are sources and sinks related to the divergence? What is the meaning of the gradient of a scalar? What is the meaning of the curl of a vector? What is the meaning of the Laplacian of a scalar? Why are complex numbers necessary? How does an exponential function with a complex argument differ from an exponential function with a real argument? Why are double and triple integral calculated? Compare and contrast algebraic equations with differential equations. What is the solution for the classical pure harmonic oscillator? Describe how underdamped, critically damped and overdamped oscillators are different. What are the requirements on matrices to be multiplied together? Name two technique used to solve systems of equations. Compare and contrast eigenvalue/eigenvector problems of matrix algebra to eigenvalue/eigenfunction problems of differential equations. Terms and Definitions Cartesian coordinates Spherical polar coordinates Cylindrical coordinates Differential volume element Complex number Modulus Phase angle Argand plane Complex conjugate Euler’s identity Vector Scalar Gradient Del operator Divergence Source Sink Curl Laplacian Ordinary differential equation Partial differential equation Homogeneous differential equation Order of a differential equation Linear differential equation Nonlinear differential equation Auxiliary equation Harmonic oscillator Hooke’s Law Force constant (spring constant) Damping (harmonic oscillator) Damping constant Overdamped Underdamped Critically damped Decay envelope Transpose of a matrix Diagonal of a matrix Determinant of a matrix Inverse of a matrix Gaussian elimination Cramer’s rule Eigenvalue Eigenvector Characteristic equation2Key Equations First-tier 2ddxdydzrsindrddτ= = θ θ φ ()()iecos isin±αθ=αθ ± αθ ˆˆˆijkxyz∂∂∂∇= + +∂∂∂ Fundamental Concepts of Quantum Mechanics Concepts What is difference between a particle and a wave? What are the consequences of the wave-particle duality? Heisenberg’s uncertainty principle Quantization of energy Quantization of angular momentum Schrödinger equation is a wave equation. Diffraction of particles Why is wave-particle duality counterintuitive? Discuss how photoelectric effect demonstrates that the photon is a particle. What is deBroglie’s relationship? What is the momentum operator? What is the kinetic energy operator? What are the requirements of the wavefunction? Why are the requirements of the wavefunction necessary? Why are wavefunctions normalized? Compare and contrast the trajectory of a classical particle with the wavefunction of a quantum particle. How does one get information from the wavefunction? What is special about an eigenfunction of an operator? What is convenient about orthonormal basis sets? What is a probability density? How is the probability density different from the wavefunction? What does one mean by the statement: the wavefunction is not real? What is the relationship between operators and observables? How are Hermitian operators related to observables? Explain how the Heisenberg uncertainty principle affects how quantum mechanical properties are measured. When operators commute, what is the consequence for measuring the observables associated with the operators? Understand the postulates of quantum mechanics as discussed in the McQuarrie text. Terms and Definitions Wave-particle duality Diffraction Single-slit diffraction Double-slit diffraction Photoelectron effect Work function Photon Planck’s constant DeBroglie waves DeBroglie wavelength3Heisenberg’s uncertainty principle Quantum mechanical state Trajectory Wavefunction Continuity of a function Operator Hermitian operator Probability density Born’s interpretation of the wavefunction Normalization Observable Expectation value Commutator Commutating operators Schrödinger equation Hamiltonian Eigenfunction Hilbert space “Spanning the space” Basis set Orthogonality Orthonormality Kronecker’s delta Key Equations First-tier h2=π ˆxx= xˆpix∂=∂ Eh=ν= ω c=νλ Eh=ν−Φ hpλ= Probability density between a and b b*ad=ψψ τ∫ ˆOˆOψψ=ψψ EEψ= ψ ≡ ψ = ψHH []ˆˆˆˆˆˆx,p xp px i=−= xp2∆∆≥ Et2∆∆≥ ab abdφφ τ=δ∫ Simple Quantum Mechanical Models Concepts How is the particle in free space different from other quantum mechanical models? Know how to write the kinetic energy of a particle in terms of its momentum. How do the requirements of the wavefunction force quantization of energy? How does the quantization of energy create a zero-point energy? How is the zero-point energy related to the Heisenberg uncertainty principle? Under what conditions can tunneling occur? How is tunneling related to the magnitude of the kinetic energy and potential energy of a particle? What are some applications of tunneling? Explain the technique of separation of variables to solve a differential equation. How are degrees of freedom related to the number of quantum numbers needed to characterize a quantum mechanical state? How does the symmetry of a potential function affect the degeneracy of energy levels? What is the potential energy operator for a harmonic oscillator?4Why is the dimensionless position used instead of position in the solution of the harmonic oscillator? What is the relationship between the frequency of vibration and the force constant? What are the three parts of the harmonic oscillator wavefunction? What is a generating function and why are they useful? Why is Dirac’s bra-ket notation used? Terms and Definitions Free space Hamiltonian Wavevector Wavenumber Particle-in-a-box Boundary conditions Quantization Zero-point energy Step potential Barrier potential Tunneling Classically forbidden region Degree of freedom Separation of variables Degeneracy Commutator Hermite polynomial Dimensionless position (momentum) Gaussian function Generating function Recursion relation Bra-ket notation Key Equations First-tier 2pˆT2m= 2ˆT2m=− 21ˆVkx2= 00xLVotherwise≤≤=∞ 1E2ν⎛⎞=ων+⎜⎟⎝⎠ kω=µ pk= 1k =λ 2ω=πν ()0Fkxx=− − ()22HONH eξ−ννψ= ξ ()vH Hermiteξ− Second-tier PIB2nxsinLLπ⎛⎞ψ=⎜⎟⎝⎠ 222PIB2nhE2mLπ= Angular Momentum Concepts Know the classical expression for the angular momentum.