Collection of formulae Quantum mechanics. Basic Formulas Division of Material Science Hans Weber. Operators

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1 Basic Formulas Division of Material Science Hans Weer The de Broglie wave length λ = h p The Schrödinger equation Hψr,t = i h t ψr,t Stationary states Hψr,t = Eψr,t Collection of formulae Quantum mechanics ψr,t = ψr φt, where φt = e iet/ h The Hamiltonian of one particle H = h m +Vr Proaility flux jr = h im ψ r ψr ψr ψ r Equation of continuity Pr,t+ jr,t =, where t Pr,t = ψr,t = ψ r,tψr,t Group velocity v g = ω k Operators Physical quantity x-representation p-representation Position coordinate x x x = i h p x Position vector r r i h p x-component of momentum p x i h x p x Momentum p i h p Kinetic Energy T = p m µ 1 µ P Potential energy Vr,t Vr,t Vi h p,t Total energy T = p m Commutator of operator A and B [A,B] = AB BA Commutator algera [AB,C] = A[B,C]+[A,C]B [A,B +C] = [A,B]+[A,C] +Vr,t h µ +Vr,t 1 µ P +Vi h p,t [A,[B,C]]+[B,[C,A]]+[C,[A,B]] = Expectation value A = ψ AψdV, where ψ is a normalized wave function. Variance A = A A = Uncertainty relation A B 1 i[a,b] A A Time dependence of expectation values d dt A = ī A h [H,A] + t The Expansion theorem Suppose that A is a Hermitian operator with discrete eigenfunctions u n and continuous eigenfunctions u q. Then the aritrary function ψx may e expanded in terms of u n and u q as ψx = c n u n + c q u q dq n

2 where if c n =< u n ψ > c q =< u q ψ > < u n u n >= 1, < u q u q >= δq q. Hermitian operators The eigenvalues of a Hermitian operator A are real, and the eigenfunctions of A corresponding to different eigenvalues are orthogonal. Angular Momentum L = r p and Spin Orital angular momentum operator L = i hr L z = i h φ Eigenfunctions and eigenvalues to angular momentum operators The eigenfunction are the spherical harmonics Y lm θ,φ L Y lm = h ll+1y lm L z Y lm = m hy lm Commutator relations of angular momentum [L x,l y ] = i hl z, with cyclic permutation [L,L z ] = [H,L z ] = [H,L ] = Ladder operators L ± = L x ±il y L ± Y l,m = h ll+1 mm±1y l,m±1 Spin Operators S x = h 1 1 General spin state χ = aχ + +χ = Scalar product χ χ = c d S y = h a a i i = c a+d S z = h 1 1 Special Solutions Tunneling in the WKB approximation T exp dx m[vx E]/ h arrier Tunneling at a potential step and a square arrier Potential step of height V and particle of mass m incident from the left of energy E, E > V. k 1 = 1 h me and k = 1 h The transmission coefficient T is T = S transmitted S incident = 4k 1k k 1 +k and R = S reflected S incident = k 1 k k 1 +k where S is proaility flux. T +R = 1 me V Potential arrier of height V and width a and particle of mass m incident from the left of energy E, E > V. T = [1+ V sinh a 4EV E ] 1 if E < V T = [1+ V sin ] 1 a 4EE V if E > V m V E = h Particle in a square potential The potential is Vx = for < x < a and infinite elsewhere. E n = n π h ma ψx = a sinnπx a

3 Density of one-particle states in a ox of volume V As a function of energy ged p = V 6π m h The Fermi energy / N h E F = V π m The Fermi velocity EF v F = m Linear harmonic oscillator Vx = 1 kx = 1 mω x / E E n = n+ 1 hω u n x = C n H n x exp x, Note: No spin Hermite polynomials n H n x H x = 1 1 H 1 x = x H x = 4x H x = 8x 1x 4 H 4 x = 16x 4 48x +1 Eigenfunctions of the harmonic oscillator n ψ n x ψ x = 1 ψ 1 x = ψ x = ψ x = 4 ψ 4 x = 5 ψ 5 x = π 1/e 1 x π 8 π 48 π 84 π 84 π 1/xe 1 x 1/4 x e 1 x 1/8 x 1xe 1 x where = mω/ h 1/16 4 x 4 48 x +1e 1 x 1/ 5 x 5 16 x +1xe 1 x where H n x = Hermite polynomial of order n, and C n = 1 n n! 1/mω π h 1/4, = h mω. The eigenfunctions u n x of the harmonic oscillator satisfy the following recursion relation xu n x = x u n x = n+1un+1 x+ nu n 1 x, n+1n+un+ x+n+1u n x+ nn 1u n x. The Hermite polynomials satisfy the following recursion relation H n+1 ξ ξh n ξ+nh n 1 ξ =

4 Hydrogenic Atoms The Schrödinger equation in spherical coordinates h 1 L m r rr + mr +Vr ψr = Eψr Spherical harmonics Y l,m θ,φ Klotytefunktioner l m Y l,m θ,φ Y, = 1 4pi 1 Y 1, = 4π cosθ 1 ±1 Y 1,±1 = 8π e±iφ sinθ 5 Y, = 16π cos θ 1 15 ±1 Y,±1 = 8π e±iφ sinθcosθ 15 ± Y,± = π e±iφ sin θ 7 Y, = 16π 5cos θ cosθ ±1 Y,±1 = π sinθ 5cos θ 1 e ±iφ ± Y,± = π sin θcosθ e ±iφ 5 ± Y,± = 64π sin θ e ±iφ Y l, m = 1 m Y l,m Radial wave functions of hydrogenic atoms n l R n,l r 1 R 1, r = / R, r = a a /e r/a 1 r a e r/a / 1 R,1 r = 1 r a a e r/a / R, r = a 1 r a + r e 7a r/a / 1 R,1 r = 4 r a a 1 r 6a e r/a / e R, r = 7 r r/a 5 a a / 4 R 4, r = 4a 1 r 4a + r 8a r 19a e r/4a Associated Laguerre polynomials L nx L 1 1x = 1 L 1 x = x 4 L x = L 1 x = x +18x 18 L x = 6x+18 L x = 6 Expectation values of r k in hydrogenic atoms r = 1 r = n [ n ll+1 ] a r 1 = 1 n a r = r = [ 5n +1 ll+1 ] a n l+1 a 1 n ll+ 1 l+1 a Energy of a diatomic molecule E = E trans +E vi +E rot = p m + hωn+ 1 h + I ll+1 Perturation Theory The Hamiltonian H = H +H 1 where H is the un pertured Hamiltonian with a known solution φ n and H 1 is the perturation. First order perturation E 1 n =< φ n H 1 φ n > ψ n 1 = φ n + < φ k H 1 φ n > E k n n Ek φ k

5 Second order perturation E n = < φ k H 1 φ n > E k n n Ek Statistical Physics The Boltzmann factor for a stat of energy E i P i exp E i /k B T The Partition function Tillståndssumman = j exp E j /k B T and hence P i = exp E i/k B T Expectation value of n for a harmonic oscillator n = 1 exp hω/k B T 1 The average energy of a quadratic degree of freedom < E >= 1 k BT