Fundamentals of Quantum Mechanics: Operators and Eigenfunctions, Study notes of Quantum Mechanics

Download Fundamentals of Quantum Mechanics: Operators and Eigenfunctions and more Study notes Quantum Mechanics in PDF only on Docsity! 157 VII. 3. APPLICATION EXAMPLES FOR THE HARMONIC OSCILLATOR A good example for a harmonic oscillator in one dimension is the HCl molecule. The hydrogen - and chlorine nucleus oscillate around their equilibrium positions. Fig. 7.5 Model for the HCl molecule (small circle H, large circle Cl ) If we denote with X0 the distance of the equilibrium positions and with X the instantaneous position, then we have as oscillator coordinate x = X - X0 . From ω 2 = + k m mH Cl we can calculate the fundamental frequency, if we know the force constant, k, and the atomic masses. Fig. 7. 6 Energy states of the harmonic oscillator Since HCl has a dipole moment , the variable dipole moment belongs to x(t) .e X+ ⋅ 0 e x t + ⋅ ( ) HCl gas absorbs light at a wavelength of 8 = 3.5 :m . The absorbed energy ST causes a transition n = 0 to n = 1 or n =1 to n = 2 of the molecular oscillator, the dipole strength of which can be calculated using (6.4.16). 158 VIII.1. THE EQUIVALENCE OF OPERATOR AND MATRIX FORMULATION OF QUANTUM MECHANICS. a) Compilation of average values 1) multiplicative operators $x x dx= ∗z ψ ψ V x V x dx( ) ( )= ∗z ψ ψ e e dx= ∗zψ ψ 2) differential operators E i t dx= − ∂ ∂ ∗zψ ψh E H dx= ∗zψ ψ$ $p i x dxx = ∂ ∂ ∗zψ ψh 3) Differential operators in position space can be multiplicative in momentum space and vice versa Position space Momentum space 161 Proof: Ri and Rj are two eigenfunctions for Ei and Ej. The time independent Schroedinger equation for and Rj , respectively areψ i ∗ (8.1.5) $ $H E H Ej j j i i iψ ψ ψ ψ= =∗ ∗ ∗ ( : hermitian operator)$H Multiplication from the left with of the first and Rj of the second equation , integration, andψ i ∗ following subtraction of the first equation from the second yields: (8.1.6) 2 3 3( ) ( ) 2 i j j i i j i j d x E E d x m ψ ψ ψ ψ ψ ψ∗ ∗ ∗   ∆ − ∆ = −    ∫ ∫ h The left side can be expressed as surface integral over the probability current j miij i j j i = ∇ − ∇∗ ∗ h 2 ( )ψ ψ ψ ψ (8.1.7)⇒ = − ∗zzzhi j df E E d xij i j i j( ) 3 ψ ψ We extend the integration over the infinite space and correspondingly over a surface , which will be infinite at the limit. We demand that R and LR disappear at infinity fast enough, i. e. at least as 1/r for R . lim r ij j df →∞ =zz 0 because j r f r product r egral for rij ~ , ~ ~ int 1 1 0 3 2 ⇒ → → ∞ We obtain 162 ( )E E d xi j i j− = ∗zψ ψ 3 0 (8.1.9) ⇒ = ⇒ ≠ ≠ ⇒ = ∗ ∗ z z E E d x E E d x i j i j i j i j ψ ψ ψ ψ 3 3 0 0 We assume that the functions are normalized (8.1.10)ψ ψ δi j ijd x ∗z =3 The eigenfunctions of form an orthonormal system, i.e. a class of functions, for which the$H above normalization condition is valid. The system of eigenfunctions is complete. Explanation: The Ri form a class cE. An arbitrary function f (x) (square integrable) within this class, cE, can be represented as linear combination g x ck k k n ( ) = = ∑ ψ 1 so that f(x) = .lim ( ) n g x →∞ Because of we haveψ ψ δi j ijd x ∗z =3 (8.1.11)f x dx c c ci ij j ij i i ( ) 2 2= =∗∑z ∑δ 163 This “Bessel’s equation” is for all a necessary and sufficient condition forf x cE( ) ∈ completeness. We can express ci by (8.1.12)c x f x dxi i= ′ ′ ′ ∗zψ ( ) ( ) and substitute in (8.1.12)f x c xi i i ( ) ( )= ∑ ψ f x x x f x dxi i i( ) ( ) ( ) ( )= ′ ′ ′∑z ∗ψ ψ or (8.1.13)f x K x x f x dx( ) ( , ) ( )= ′ ′ ′z where K x x x x x xi i i ( , ) ( ) ( ) ( )′ = − ′ = ′∗∑δ ψ ψ d) Expansion of R (x,t) in a complete orthonormal system All eigenfunctions of (the class of functions cE ) form a complete orthonormal $H system. Then we can expand (8.1.14)ψ ψ φ= =∑ ∑ − c ck k k k k i E t k k exp h R is normalized 166 sign. SYMMETRIC : ~ T T= ANTISYMMETRIC : ~ T T= − Note that the transpose of a COLUMN matrix is a ROW matrix . ~ ( , ,........., )a a a an= 1 2 COMPLEX CONJUGATE of a matrix, T* T T T T T T T T T T n n n n nn ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ = F H GGGG I K JJJJ 11 12 1 21 22 2 1 2 . . . . . . . A matrix is REAL , if all its elements are real, and IMAGINARY, if they are all imaginary. REAL : T* = T IMAGINARY : T* = - T The HERMITIAN CONJUGATE (or ADJOINT) of a matrix (indicated by a dagger : T ^ ) is the transposed conjugate T ^ = ∗~T A square matrix is HERMITIAN (or SELF-ADJOINT), if it is equal to its Hermitian conjugate; if the Hermitian conjugation introduces a minus sign, the matrix is SKEW HERMITIAN (or ANTI-HERMITIAN) HERMITIAN : T ^ = T SKEW HERMITIAN: T ^ = - T 167 (8.1.20) F F F F F F F F F F F F F F F F F F 11 12 13 21 22 23 31 32 33 11 21 3 12 22 32 13 23 33 . . . . . . . . . . . . . . `F H GGGG I K JJJJ = F H GGGG I K JJJJ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ In the same sense one talks about the hermiticity of an operator, , which generates this matrix.$F If the operator, , is hermitian one has$F ψ ψ ψ ψ∗ ∗z z=$ ( $ )F dx F dx or (8.1.21)ψ ψ ψ ψ ψ ψi j i j i jF dx F dx F dx ∗ ∗ ∗ ∗= =zz z( $ ) ( $ ) ( $ ) The proof of this relation is based on the expansion of the first integrals and comparison of the terms with the same Tij . e) Derivation of an operator with respect to position and momentum We ask ourselves, what is the result, if the momentum operator operates on an arbitrary operator .$F $p i x = ∂ ∂ h (8.1.22)$ $ $ $pF i x F i F x ψ ψ ψ= ∂ ∂ F HG I KJ + ∂ ∂ h h 168 (8.1.23)⇒ ∂ ∂ = − h i x F pF Fp$ $ $ $ $ Correspondingly $x i p i p = ∂ ∂ = − ∂ ∂ h h (8.1.24) h i p F Fx xF x F ∂ ∂ = − = −$ $ $ $ $ $, $ (8.1.25) d dt F t t F t dx t F t dx F t t dx kl k l k l k l ( ) $ ( ) $ ( ) $ ( ) = ∂ ∂ F HG I KJ + ∂ ∂ F HG I KJ + ∂ ∂ F HG I KJ ∗ ∗ ∗ z zz ψ ψ ψ ψ ψ ψ Substitution of the Schroedinger equation into (8.1.25) yields (8.1.26) d dt F t d dt F t dx i HF FH dx t F t dx kl k l k l k l ( ) $ ( ) ( $ $ $ $ ) $ ( ) = = − + ∂ ∂ ∗ ∗ ∗ z z z ψ ψ ψ ψ ψ ψh 171 If the Nk are not also eigenfunctions of , we have terms of the form$F (8.1.29)F ekl i E E tk lh ( )− Otherwise we would have only Fkk . do not commute or⇒ $ $H and F (8.1.30) d dt F i HF FHkl kl= −h ( $ $ $ $ ) One could assume, that in the case of non-stationary solutions R (x,t) , the factors ck would be functions of time: ck(t) This is not the case, if the operator does not contain the time$H explicitly [via V(x,t)] Proof: with ψ ψ= ∑ c t x tk k k ( ) ( , ) ∂ ∂ = − t i Hk kψ ψh $ − ∂ ∂ = FHG I KJ + F HG I KJ RST UVW − F HG I KJ∑ ∑ h h i t d dt c c d dt ikk k k k k ψ ψ ψ $H c Ek k k k ψ ψ= ∑ 172 − ∂ ∂ = − FHG I KJ − F HG I KJ =∑ ∑ ∑ h h h i t i d dt c i c d dt c Ek k k k k k k k k k ψ ψ ψ ψ 0 = FHG I KJ + F HG I KJ ∑ +∑ ∑ ∑ − − ∂ ∂ = h h 1 244 344 h i d dt c i c d dt c Ek k k k k k c E because of i t H k k k k k k k k k k ψ ψ ψ ψ ψ ψ$ 0 = FHG I KJ ⋅∑ ∗d dt c and egratek k k lψ ψ int 0 = FHG I KJ = F HG I KJz∑ ∑ z∗ ∗ddt c dx ddt c dxkk l k kk l kψ ψ ψ ψ 0 0= FHG I KJ ⇒ =∑ d dt c d dt c tk k lk kδ ( ) cl does not depend on time. Q.e.d. What does it mean, if commute ?$ $H and F means, that the state Rk has a defined energy value Ek $H Ek k kψ ψ= 173 $ $ $FH E F E fk k k k k kψ ψ ψ= = $ $ $ $HF Hf f H f E E fk k k k k k k k k k kψ ψ ψ ψ ψ= = = = Rk is simultaneously an eigenfunction of . In other words, since the eigenvalues fk and$ $H and F Ek form diagonal matrices, (fkk) and (Ekk) commute with each other. Ek means measurement of a defined Ek , if we have the pure state described by Rk . But simultaneously fk can be measured. (8.1.32)⇒ − =$ $ $ $HF FH 0 means possibility to measure Ek and fk simultaneously with arbitrary accuracy. The commutation relation is valid now only for Rk , the eigenfunctions of the energy operator. Since we can expand each arbitrary function R in terms of Rk ψ ψ= ∑ck k k it is generally valid. Classical analog : We expect that is the quantum analog d dt F FH HF i = −$ $ $ $ h I have reversed here the order because of the factor i in the denominator. With - i in the nominator we would get − − i HF FHh $ $ $ $ of one of Hamilton’s equations, 176 Fig. 8. 9 Example of a core potential (8.1.34)ψ ψ φ= +∑ z −c dEc x ek k k E E i E t ( ) h NE(x) eigenfunctions for a continuous eigenvalue E g) Addition and multiplication of matrices Addition: (A) + (B) = (C) (A + B)ik = Aik + Bik = Cik Multiplication of matrices (a) multiplication with a constant c (Aik) = (cAik) (b) multiplication of two matrices 177 (8.1.36)AB A B ik ij jk j b g = ∑ i is row index, k is column index of product matrix (Scheme)− − − − − − − • U V || W || row column 6 74 844 | | | | (8.1.37)A x xij j j i′ =∑ B x xjk k k j′′= ′∑ (8.1.38A B x x C xij jk k i ik k kjk ′′ = = ′′∑∑ Examples for multiplication A A A A B B B B A B A B A B A B A B A B A B A B 11 12 21 22 11 12 21 22 11 11 12 21 11 12 12 22 21 11 22 21 21 12 22 22 F HG I KJ ⋅ F HG I KJ = + + + + F HG I KJ ASSOCIATIVE LAW A (BC) = (AB) C In general the commutative law is not valid AB Ö BA Example: AijBjk = Cik AB = C ; BkjAji = Cki transpose of C ~~ ~ BA C= (The ' symbol is often omitted; the convention is then that the sum is over the indices in the product, which are the same; in the above case, therefore , j ) The inverse matrix We substitute into (8.1.38) ′′ =x x 178 ⇒ = =A B x x I xij jk k i ii i I : Unit matrix AB = I ∫ B = A-1 (Inverse matrix) A A-1 = A-1A = I One finds the elements of A-1 by solving a system of linear equations Det|Aij| : determinat of the matrix ( Aij ) det|Aij| : subdeterminant of element Aij Det A A A A A A A A F A ij n n n n nn = 11 12 1 21 22 2 1 2 . . . . . . . det . [ ] . . [ ] . . . . [.] . . [ ] [ ] [.] [ ] [.] . . . [.] . . . [ ] . A A A A A A A A A A A A A A A A A ij i j j n j n i i ij in n n nj nn = − +1 11 12 1 1 21 22 2 2 1 2 1 2 b g erase the elements in [ ] , i.e. the ith row and jth column