Atkins & de Paula: Atkins Physical Chemistry 9e Checklist of key ideas. Chapter 8: Quantum Theory: Techniques and Applications

Transcription

1 Atkins & de Paula: Atkins Physical Chemistry 9e Checklist of key ideas Chapter 8: Quantum Theory: Techniques and Applications

2 TRANSLATIONAL MOTION wavefunction of free particle, ψ k = Ae ikx + Be ikx, E = k 2 ħ 2 / m. k 2 k continuous variable 8.1 Particle in a box boundary conditions, the constraints on a wavefunction at points in space. wavefunctions, ψ n (x) = (2/L) 1/2 sin(nπx/l). energies, E n = n 2 h 2 /8mL 2, n = 1,2,... quantum number, an integer (and in some cases a half-integer) that labels the state of a system. zero-point energy, the lowest possible energy; E 1 = h 2 /8mL 2.

3 8.1 Particle in a box (cont..) correspondence principle: classical mechanics emerges from quantum mechanics at high quantum numbers. orthogonality: two wavefunctions are orthogonal if ψ i * ψ j dτ = 0. orthonormal, functions that are both normalized and mutually orthogonal. 8.2 Motion in two or more dimensions. separation of variables, the possibility of writing ψ(x,y) = X(x)Y(y). wavefunctions, ψ n_1n_2 (x,y) = {2/(L 1 L 2 ) 1/2 }sin(n 1 πx/l 1 ) sin(n 2 πy/l 2 ).

4 Chapter 9: Quantum Theory: Techniques and Applications 8.2 Motion in two or more dimensions (cont..) energies, E n_1,n_2 = (n 1 2 /L n 2 2 /L 2 2 )h 2 /8m. degenerate wavefunctions, different wavefunctions corresponding to the same energy. 8.3 Tunnelling tunnelling, penetration into or through classically forbidden regions. transmission probability, the ratio of the probabilities of transmission and incidence; for a rectangular barrier: κl κl ( e e ) 1 2 2κL T = ε ( 1 ε ) e, ε = E/V. 16ε ( 1 ε )

5 VIBRATIONAL MOTION harmonic motion, motion when the restoring force is proportional to the displacement; F = kx. force constant, the constant k in Hooke s law of force, F = kx. parabolic potential energy, a potential energy of the form ½kx The energy levels (of a harmonic oscillator) energy levels, E v = (v + ½)ħω, ω = (k/m) 1/2, v = 0,1,2... zero-point energy: E 0 = ½ħω. separation of neighbours: ħω for all v.

6 8.5 The wavefunctions (of a harmonic oscillator) wavefunctions, ψ v = N v H v (y)e y^2/2, y = x/α, α = (ħ 2 /mk) 1/4. Hermite polynomial, a type of orthogonal polynomial. mean displacement, x = 0. mean square displacement, x 2 = (v + ½)ħ/(mk) 1/2. virial theorem: if the potential energy of a particle has the form V = ax b, then its mean kinetic and potential energies are related by 2 E K = b V.

7 ROTATIONAL MOTION 8.6 Rotation in two dimensions: the particle on a ring angular momentum, J z, the moment of momentum; J z = ±pr. moment of inertia, for a point mass m at a distance r from the axis of rotation, I = mr 2. cyclic boundary conditions, the requirement that ψ(φ + 2π) = ψ(φ). wavefunctions, ψ m_l (φ) = (1/2π) 1/2 e im_lφ, m l = 0, ±1, ±2,... energies, E m_l = m l 2 ħ 2 /2I, m l = 0, ±1, ±2,... zero-point energy: E 0 = 0.

8 8.6 Rotation in two dimensions: the particle on a ring (cont..) orbital angular momentum operator, l z = xp y yp x = (ħ/i) / φ. z-component of angular momentum, l z = m l ħ. 8.7 Rotation in three dimensions: the particle on a sphere laplacian, the operator 2 = 2 / x / y / z 2. Schrödinger equation, (ħ 2 /2m) 2 ψ = Eψ. legendrian, the operator Λ 2 = (1/sin 2 θ) 2 / φ 2 + (1/sin θ)( / θ)sin θ ( / θ). orbital angular momentum quantum number, l = 0, 1, 2,....

9 8.7 Rotation in three dimensions: the particle on a sphere (cont..) spherical harmonics, the functions Y l,m_l (θ,φ). energies, E l,m_l = l(l + 1) ħ 2 /2I, l = 0,1,2,..., independent of m l. degeneracy of energy level with quantum number l is 2l + 1. magnitude of angular momentum, J = l(l + 1) 1/2 ħ. z-component of angular momentum, m l ħ, m l = l, l 1,..., l. space quantization, the restriction of a rotating body to (2l + 1) discrete orientations. vector model, a pictorial representation of angular momentum as cones of length l(l + 1) 1/2 and vertical height m l units.

10 8.7 Rotation in three dimensions: the particle on a sphere (cont..) exciton, an electron hole pair in a crystal. quantum dot, a three-dimensional nanocrystal of semiconducting material consisting of atoms. 8.8 Spin spin, an intrinsic angular momentum of a particle. spin quantum number, s, a quantum number that species the spin angular momentum through {s(s + 1)} 1/2 ħ; for an electron, s = ½. spin magnetic quantum number, m s, a quantum number that specifies the z-component of the spin angular momentum through m s ħ; for an electron, m s = ±½.

11 8.8 Spin (cont..) Stern Gerlach experiment, the detection of electron spin by the deflection of an electron beam in an inhomogeneous magnetic field. fermion, a particle with half-integral spin quantum number. boson, a particle with integral spin quantum number. TECHNIQUES OF APPROXIMATION 8.9 Time-independent perturbation theory perturbation theory, a technique for taking into account the influence of a small addition to a simple hamiltonian operator.

12 8.9 Time-independent perturbation theory (cont..) time-independent perturbation theory, perturbation theory in which the perturbation does not vary with time. matrix element, a quantity of the form Ω = ψ * Ωˆ ψ dτ nm n m. time-dependent perturbation theory, perturbation theory in which the perturbation or its influence does vary with time. transition dipole moment, the quantity µ = ψ * ˆ µ ψ dτ ; a transition intensity is z,fi f z i proportional to the square modulus of the transition dipole moment.