Wave Mechanics. January 16, 2017

Transcription

1 Wave Mechanics January 6, 7 The ie-dependen Schrödinger equaion We have seen how he ie-dependen Schrodinger equaion, Ψ + Ψ i Ψ follows as a non-relaivisic version of he Klein-Gordon equaion. In wave echanics, he funcion Ψ is called he wave funcion, alhough we will shorly see ha is generalizaion is called he sae of he syse. Nex, we look a is inerpreaion and soluion.. Conservaion of ψ ψ There is a conservaion law associaed wih he Schruodinger equaion. Muliplying eq. by he coplex conjugae wave funcion, Ψ, we have and he coplex conjugae of his is Taking he difference of hese gives Now define Ψ Ψ + Ψ Ψ i Ψ Ψ Ψ Ψ + Ψ Ψ i Ψ Ψ Ψ Ψ + Ψ Ψ + Ψ Ψ Ψ Ψ i Ψ Ψ + i Ψ Ψ Ψ Ψ Ψ Ψ i Ψ Ψ i Ψ Ψ Ψ Ψ Ψ Ψ so ha ρ Ψ Ψ J i Ψ Ψ Ψ Ψ ρ + J

2 This is he coninuiy equaion, and i shows ha he inegral of ρ over a volue is conserved o he exen ha no curren J flows across he boundary of. Specifically, consider d ρ x, d 3 ρ x d d3 x Jd 3 x n Jd x We see ha any change in ρ x, is exacly given by he flux of he curren J across he boundary. We inerpre his inegral as he probabiliy of finding he paricle described by ρ in he volue, giving ρ Ψ Ψ he inerpreaion of a probabiliy densiy. Furher jusificaion for his inerpreaion is given below. Since he wave funcion Ψ vanishes a infiniy, he inegral of ρ Ψ Ψ over all space is sricly consan. This represens he probabiliy of finding he paricle soewhere. In keeping wih he requirens of probabiliy, we agree o noralize he wave funcion so ha his inegral is one: ρ x, d 3 x Ψ Ψd 3 x all space all space. Separaion of he ie variable and superposiion The os coon approach o a soluion, which works when he poenial depends on posiion only, is separaion of he ie variable. Le he wave funcion, Ψ x, be wrien as a produc, S Ψ x, ψ x T Subsiuing, we have dt ψ x + x ψ x T i ψ x d so dividing by Ψ we have ψ x ψ x + x ψ x i dt T d Since he lef side depends only on x and he righ only on, each side us equal soe consan, which using soe foresigh we call E. Then we have wo equaions, ψ E x + x ψ E x Eψ E x i dt ET d The firs is he saionary sae Schrödinger equaion, while he second is iediaely inegraed o give T e i E There is no obvious resricion on he separaion consan, E, bu here will be in bound saes: no every value of E will be consisen wih he boundary condiions. We will see his effec shorly. Suppose all values of he energy are allowed. Then for each value of E, we have a soluion of he for, Ψ E x, ψ E E, x e i E

3 Since he ie-dependen Schrödinger equaion is linear, any superposiion of hese is allowed. The general soluion is an arbirary linear cobinaion of hese paricular ones. Inroducing arbirary apliudes A E for each energy of wave funcion, we have he general soluion, Ψ x, A E ψ E E, x e i E de 3 Noice ha while each Ψ E oscillaes wih he single frequency ω E, in he superposiion he ie behavior ay becoe quie coplex. When we deal wih bound saes, so ha he wave funcion us saisfy boundary condiions a a finie disance, he allowed energies will be discree. Le n label he sae wih energy E n, so we ay wrie he saionary sae soluion as ψ n x. Now he soluion is a linear cobinaion over all n, Ψ x, where again, he consans A n are arbirary. The ie-independen Schrödinger equaion A n ψ n E n, x e i En 4 n Once we have separaed he ie dependence, we sill have a differenial equaion for he spaial dependence of he wave funcion. The reaining equaion is he saionary sae Schrödinger equaion, ψ E x + x ψ E x Eψ E x 5 The nae saionary sae refers o he fac ha soluions for a single eigenvalue E have rivial ie dependence, Ψ Aψ E x e i E wih a probabiliy densiy ha is saionary, ρ x, Ψ Ψ A ψ E x ψ E x The presence of he failiar Laplacian ells us o expec unique soluions once we ipose boundary condiions. A linear differenial operaor L acing on a funcion which reurn a consan α ies he funcion, Lf αf is called an eigenvalue equaion, and he consan α is called he eigenvalue. Eq.5 has his for where he linear differenial operaor is he Hailonian in operaor for and he energy E is he eigenvalue. In general, we wrie operaors wih a ha, so he Hailonian operaor is and he ie-dependen Schrödinger equaion is Ĥ + x ĤΨ i Ψ The saionary sae Schrödinger equaion is he eigenvalue equaion ĤΨ EΨ 3

4 . Boundary condiions Suppose we have a boundary a x on a boundary surface S. Inegrae eq.5 fro an infiniesial disance ε on one side of he boundary o a disance ε on he oher side, wih n he uni noral o he boundary. Seing x +εn x εn x +εn x εn x +εn x εn ψ E + x ψ E Eψ E dε + ψ E + x ψ E Eψ E dε ψ E ε ψ E + x ψ E Eψ E dε ψe ε x + εn ψ E ε x εn + ψ E x + x ψ E x Eψ E x ε and aking he ε lii shows ha ψ E ε + ψ E ψ li E ε + ε ε x + εn ψ E ψ li E ε ε ε x εn ψ E ε boundary. Reurning o he general expression and inegraing again, x +εn x εn [ ψe ε ψ E x + εn ψ E x εn + so ha he firs derivaive us be coninuous across he x + ε n ψ ] E ε x ε n + ψ E x + x ψ E x Eψ E x ε dε ψ E x + x ψ E x Eψ E x ε Again aking ε, we see ha ψ + x ψ x. Therefore, a a boundary boh he wave funcion and is firs derivaive us be coninuous: ψ + x ψ x ψ E ψ E 6 ε + ε There is an excepion o his if he poenial diverges a he boundary, since hen he poenial er in he firs lii ay be finie, > li ε x ψ E x > ε In such cases, he firs derivaive ay have a disconinuiy. This happens wih infinie square well poenials and wih dela funcion poenials. The addiional consrain needed o deerine he soluion is generally provided by he vanishig of he wave funcion beyond he infinie barrier. While hese exaples are obviously idealizaions, hey ofen reveal cerain quanu properies correcly, and in a sipler conex. 4

5 3 acuu soluion in one diension We ay use he boundary condiions o consruc a wide range of quanu echanical soluions. exaple, any piecewise consan poenial is easily handled. In -diension, eq.5 reduces o For d ψ E x dx + x ψ E x Eψ E x We will consider several one-diensional soluions; here we explore a free Gaussian wave packe in deail. 3. Free paricle 3.. Plane waves If here is no poenial, he Schrödinger equaion reduces o d ψ E x dx This has siple oscillaory for. If we define k + Eψ E x E hen we have wih soluions ψ E d ψ E dx + k ψ E A e ikx + B e ikx π π where A E and B E are arbirary and he π facor is chosen for laer convenience. When we ipose boundary condiions, soe linear cobinaions of hese will be ruled ou, bu he appropriae condiions depend on our inerpreaion of hese soluions. To see wha he soluions ean, we look a he ie-dependen eigenfuncions Ψ E x, Ae i kx E + Be i kx+e We can exrac he energy and oenu associaed wih his sae by acing wih he energy and oenu operaors. Consider he righ oving wave given by seing B. We have ĤΨ E x, i Ψ E x, EΨ E x, pψ E x, i d dx Ψ E x, kψ E x, To exrac jus he eigenvalues, uliply by Ψ and inegrae, Ψ EĤΨ Edx E Ψ EΨ E dx E ρdx The inegral on he righ is conserved hroughou he evoluion. If we ake he volue o include all space available o he paricle, hen here can be no flux across he boundary and he inegral is consan. 5

6 These plane wave soluions are no noralizable over he whole real line. noralizaion. Noralized on a box of lengh L, Insead, we ay use a box Then Siilarly, L A e i kx E e i kx E dx A L A L L L Ψ EĤΨ Edx E Ψ E pψ E dx k so we recover he de Broglie and Planck relaions for a plane wave e ikx ω e i px E. These inegrals are called expecaion values. Since ρ is conserved, we ay evaluae i a any ie. Quie generally, choosing he iniial ie, we see ha Ψ Ψdx ψ ψdx so noralizing he saionary sae soluion is sufficien o noralize he full ie-dependen soluion. 3.. Superposiion As noed above, he plane wave soluion canno saisfy he noralizaion condiion, since Ψ EΨ E dx e i px E e i px E dx which diverges if is an unbounded region. This leads us o an addiional condiion. We will resric our aenion o hose soluions of he Schrödinger equaion which are square inegrable, eaning ha Ψ E Ψ Edx is bounded. The su of any linear cobinaion of a finie nuber of square inegrable funcions is also square inegrable, as well as a well-defined se of infinie cobinaions. Reurning our aenion o he general soluion for a free paricle, we display a class of square-inegrable soluions. Firs, noice ha since E k over all k raher han posiive E, We now need only choose A k so ha ψ ψdx ψ x p > we can label saes by k insead of E, and inegrae π 6 A k e ikx dk

7 π π dx dq dk Therefore, any funcion A k saisfying gives an allowed superposiion sae. Exaple: Gaussian wave packe A k Ae k k 4σ dk A k A q e iqx dq dka q A k dqa k A q δ q k dxe ik qx A k e +ikx dk dk A k 7 Consider a Gaussian superposiion in oenu space since p k, for any consans A, σ. Noralizing his, we require To inegrae he Gaussian, le y k k σ hen defining I e y dy consider I A so we have dk A k e k k σ dk e k k σ dk σ I π e y dy e r rdθdr π e r rdr e y dy e z dz where we conver he inegral over he yz plane o polar coordinaes. Then leing χ r, he inegral is rivial I π e χ dχ π 7

8 so ha I π. Reurning o he noralizaion, e k k σ dk πσ and we choose A. Wih his choice A k saisfies he noralizaion condiion, eq.7, guaraneeing ha he corresponding wave funcion is noralized. πσ /4 The noralized sae iself is herefore ψ x π πσ e /4 k k 4σ e ikx dk To see he spaial for of wave funcion, we carry ou his Gaussian inegral. Expanding he exponen, k k ψ x π πσ e 4σ /4 e ikx dk π πσ /4 π πσ /4 we coplee he square, 4σ k σ k ix k + 4σ k where we have defined a new inegraion variable The wave funcion now becoes ψ x exp k 4σ kk + k + ikx dk exp 4σ k σ k ix k + 4σ k dk σ k σ σ k ix σ σ k ix + + y σ σ k ix 4σ k y 4σ k + ik x + σ x + y + ik x + σ x y σ k σ dy σ dk π πσ /4 π σ πσ e ikx σ /4 σ πσ e ikx σ /4 σ k ix 4σ k exp y + ik x + σ x σdy 8 x x e y dy 4σ k

9 Therefore, σ /4 ψ x e σ x e ikx π which is siply an oscillaion wih wave nuber k in a Gaussian envelope Tie evoluion of he free Gaussian wave packe The Gaussian wave packe is a soluion o he saionary sae Schrödinger equaion. To have a full iedependen soluion we need o uliply each plane-wave ode by he corresponding energy phase, e i E. where E k k Ψ x, π A k e ikx e i Ek dk. Subsiuing his, and he noralized Gaussian apliude A k k k Ψ x, exp 8π 3 σ /4 4σ ikx + i k dk To perfor he wave vecor inegral, we coplee he square in he odified exponen, k k 4σ ikx + i k 4σ + i k k σ + ix k + k 4σ 4σ + i k k 4σ + i σ + ix 4σ y σ k + i σ σ + ix + k 4σ y k + i σ 4σ + ik x σ x k 4σ + i σ y + i σ ik x σ x i k where Le E k Then y 4σ + i k 4σ + i and p k and define he ie-dependen widh + i σ σ Ψ x, 8π 3 σ exp /4 + i σ σ exp π /4 + σ i /4 x πσ e k σ + ix ik x σ x i k + i σ i exp exp 9 ik x k p x E σ e y σ x 4σ dy + i + i k σ + ix + k 4σ

10 To see how he posiion evolves, we us look a he probabiliy densiy. A grea deal of he ie dependence is hidden in he widh, Tie evoluion of he probabiliy densiy Copue he probabiliy densiy, Then Ψ Ψ For he exponenials, we have he phase, x x + i σ p x E / πσ e x e x i exp σ σ + i σ p x E σ exp i + 4 σ 4 x + + x + + σ σ Therefore, he probabiliy densiy evolves as a Gaussian, Ψ σ Ψ + 4 σ 4 π x + x + p x p σ x p / exp [ σ x p + 4 σ 4 p x E σ i σ p x E i σ p x E i σ p x E 4i σ The apliude decreases in ie while he widh of he Gaussian increases. Meanwhile, he cener of he Gaussian oves o he righ wih velociy v p. ]