Harmonic oscillator in quantum mechanics

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1 Harmonic oscillaor in quanum mechanics PHYS400, Deparmen of Physics, Universiy of onnecicu hp:// Las modified: May, 05 Dimensionless Schrödinger s equaion in quanum mechanics a harmonic oscillaor wih mass m and frequency ω is described by he following Schrödinger s equaion: h d ψ m dx + mω x ψ(x) = Eψ(x). () The soluion of Eq. () supply boh he energy specrum of he oscillaor E = E n and is wave funcion, ψ = ψ n (x); ψ(x) is a probabiliy densiy o find he oscillaor a he posiion x. Since he probabiliy o find he oscillaor somewhere is one, ψ(x) dx =. () As a firs sep in solving Eq. () we swich o dimensionless unis: hω has he dimension of energy, hence Ē is dimensionless. Hence we inroduce he parameer, ε, hω ε E hω. (3) We divide Eq. () by hω: h d ψ mω dx + mω h x ψ(x) = εψ(x). (4) The only dimensional parameer combinaion remaining in Eq. (4), mω, has he dimension of h [lengh]. Therefore, he new variable u, mω u h x (5) is dimensionless. mω h x = u, h d mω dx = d du. (6)

2 Physics 400 Quanum harmonic oscillaor Spring 05 or d ψ du + u ψ(x) = εψ(x), (7) d ψ du + ( ε u ) ψ = 0. (8) Asympoics of he wave funcion as u ± How ψ(u) behave when u ±? Le s search for he soluion of Eq. (8) in he following form: where S(u) is a new unknown funcion. d e S du = S e S, ψ = e S, (9) d e S du = S e S + S e S, (0) where S = ds. Subsiuing Eq. (0) ino Eq. (8), arrive a he following nonlinear differenial du equaion: S + S + ε u = 0. () In he limi u ±, ε u. In addiion, as we see shorly, Therefore, Eq. () can be simplified o S S. () S u = 0 S = u S = u. (3) The choice of minus sign in Eq. (3) is he only one consisen wih he requiremen Eq. (). The soluion Eq. (3) is also consisen wih he assumpion Eq. (). Thus, lim u ± u ψ(u) = e. (4) Hermie differenial equaion Based on he resul Eq. (4) we are going o search for a soluion of Eq. (8) in he following form: ψ(u) = v(u) e u. (5) ψ = v e u v u e u (6) ψ = v e u v u e u v e u + v u e u. (7) Subsiuing Eqs. (5), (7) ino Eq. (8) and simplifying, we arrive a he following equaion: v uv + (ε )v = 0. (8) Page of 5

3 Physics 400 Quanum harmonic oscillaor Spring 05 For he laer convenience, we inroduce he noaion ε n. (9) The equaion is called Hermie equaion. v uv + nv = 0 (0) Soluions of Hermie equaion Le s search for he soluion of Hermie equaion in he following definie inegral form, v(u) = e u Y () d, () where he conour inegral in he complex plane is aken over ye unspecified conour and Y () is a ye unknown funcion. The derivaives of v(u), Eq. (), are as following: v = e u Y () d, () v = e u Y () d. (3) uv = Y () (ue u d) = Y () de u = Y () e u B A e u d ( Y ()), (4) d where A and B denoe he end poins of he conour. Le s impose he following resricion on he conour : Y () e u B A = 0. (5) In his case, uv = e u d ( Y ()). (6) d Subsiuing Eqs. (), (3), and (6) ino Eq. (0): e ( u d ) d ( Y ()) + ( + n)y () d = 0. (7) Page 3 of 5

4 Physics 400 Quanum harmonic oscillaor Spring 05 Hence, ( ) d d ( Y ) + + n Y = 0. (8) Equaion (8) can be inegraed as following. To simplify noaions, le s inroduce he following noaion: Z() = Y (), Y () = Z(). (9) Separaing variables in Eq. (8), ( dz d + + n ) Z = 0 dz ( Z = + n Finally, and Z() = n e 4 Y () = v(u) = 4 d, ψ(u) = e u eu n+ ) d ln Z = 4 n+ e n ln. (30) 4, (3) eu n+ 4 d. (3) Le s accep, for now wihou a proof, ha Eq. (3) describes a physically accepable, normalizable per Eq. (), wave funcion only if n is a non-negaive ineger. In paricular ha means ha he energy specrum of a harmonic oscillaor E = hω ( ε = hω n + ), (33) where we used Eqs. (3) and (9). If n is an ineger we can chose an arbirary closed conour ha encircles = 0 as he inegraion conour in Eq. (3). Hermie polynomials Le s have a closer look a v(u). v(u) = 4 d = eu n+ Inroducing a new inegraion variable, z, eu n+ 4 u +u d = e u e (u n+ ) d. (34) z = u, (35) and dropping an irrelevan facor, obain: v(u) = e u e z dz, (36) (z u) n+ Page 4 of 5

5 Physics 400 Quanum harmonic oscillaor Spring 05 where we abandoned an irrelevan consan and where is an arbirary closed conour encircling he poin z = u. Using auchy s formula for derivaives of analyic funcions, d k f(u) = k! du k πi he expression Eq. (36) can be rewrien as following: v(u) = e u f(z) dz, (z u) k+ dn du n e u. (37) We can see ha v(u) is acually a polynomial. The firs few non-normalized wave funcions are as following: n v n (u) ψ n (u) 0 e u u ue u 4u (4u )e u 3 8u 3 + u ( 8u 3 + u)e u Hermie s polynomials define wih a facor of ( ) n o keep posiive he coefficien nex o he highes power of he argumen: H n (u) = ( ) n e u dn du n e u. (38) For he reference, he explici expression for Hermie polynomials is as following: H n (x) = n m=0 ( ) m n m n! m! (n m)! xn m. (39) A wave funcion in quanum mechanics defined up o an arbirary consan, hence he wave funcion of a harmonic oscillaor can be expressed as following: ψ n (u) = e u Hn (u). (40) References [] Lev D. Landau and Evgeny M. Lifshiz. Quanum Mechanics Non-Relaivisic Theory, volume III of ourse of Theoreical Physics. Buerworh-Heinemann, 3 ediion, 98. Page 5 of 5