8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology. Lecture 6. Time Evolution and the Schrödinger Equation

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1 8.04: Quatum Mechaics Proessor Alla Adams Massachusetts Istitute o Techology 2013 February 26 Lecture 6 Time Evolutio ad the Schrödiger Equatio Assiged Readig: E&R 3 all, 5 1,3,4,6 Li , Ga. 2 all =4 Sh. 3, 4 The Schrödiger equatio is a partial dieretial equatio. For istace, i the the Schrödiger equatio becomes pˆ2 ˆ mω 2 xˆ2 E = +, 2m 2 ψ 2 2 ψ mω 2 x 2 i = + ψ. t 2m x 2 2 ˆ O course, E depeds o the system, ad the Schrödiger equatio chages accordigly. ˆ To ully solve this or a give E, there are a ew dieret methods available, ad those are through brute orce, extreme cleveress, ad umerical calculatio. A elegat way that helps i all cases though makes use o superpositio. Suppose that at t = 0, our system is i a state o deiite eergy. This meas that where ψ(x, 0; E) = φ(x; E), Êφ(x; E) = E φ(x; E). Evolvig it i time meas that ψ E i = E ψ E. t Give the iitial coditio, this is easily solved to be through the de Broglie relatio Note that ψ(x, t; E) = e iωt φ(x; E) E = ω. p(x, t) = ψ(x, t; E) 2 = φ(x; E) 2,

2 2 8.04: Lecture 6 so the time evolutio disappears rom the probability desity! That is why waveuctios correspodig to states o deiite eergy are also called statioary states. So are all systems i statioary states? Well, probabilities geerally evolve i time, so that caot be. The are ay systems i statioary states? Well, othig is eteral, ad like the plae wave, the statioary state is oly a approximatio. So why do we eve bother with this? The aswer lies i superpositio! I solves the Schrödiger equatio, the so does iω ψ t (x, t) = e φ (x) iω ψ(x, t) = c t ψ (x, t) = c e φ (x) (0.1) thaks to the liearity o that equatio. This meas that ay ψ(x) at ay time satisyig appropriate boudary coditios ca be expressed as a superpositio o eergy eigeuctios. But what do these eergy eigestates look like ayway? The aswer depeds o the particular orm o E, ˆ which depeds o the system i questio. For example, let us cosider a ree particle. This meas that V (x) = 0, so pˆ2 2 2 Ê = =. 2m 2m x 2 We eed to solve or the eigeuctios give by so Makig the substitutio Êφ E = E φ E, 2 2 φ E = E φ E. 2m x 2 k 2 2mE yields the geeral solutio φ(x; E) = αe ikx + βe ikx where α ad β are complex costat coeiciets satisyig ormalizatio. 2

3 8.04: Lecture 6 3 Aother way to get this would be through the kowledge that the eigeuctios o ˆp are e ikx ad e ikx. Note that 2 k 2 E = 2m ikx is the eergy o both the states e ad e ikx. This is a example o a degeeracy: sometimes, dieret states happe to share the same eigevalue or a particular observable operator. Take ote o the ormalizatio: multiplyig a eigeuctio by a costat leaves it still as a eigeuctio. We wat to ix the ormalizatio such that (φ E φ E ' ) = δ(e E ' ). I the previous case, it makes sese to choose the ormalizatio (φ k φ k ' ) = δ(k k ' ) as k is cotiuous, so Cotiuig with that example, φ(x; k) = 1 e ikx. 2π 2 k 2 E = 2m implies that k 2 ω =, 2m so the solutio or all time is a travelig wave i(kx ωt) ψ(x, t; E) = e disregardig ormalizatio. Note that the phase velocity ω k v p = = k 2m is hal o the classical velocity o a ree particle, while the group velocity ω k v g = = k m is exactly the classical velocity. I geeral, the group velocity is more represetative o the classical velocity tha is the phase velocity, as the group velocity is observable while the phase velocity is ot. Let us move o to a example o a otrivial potetial. The iiite square well, also kow as the particle i a box, is a idealizatio o a large, deep potetial. It is give by V (x) = {0 or 0 x, otherwise}.

4 4 8.04: Lecture 6 The implicatio o this is that P(x > ) = P(x < 0) = 0, implyig that ψ(x) = 0 outside o the box. Iside, we ca use the eergy eigevalue equatio ad deie so that the solutios are The boudary coditios imply B = 0 ad 2 2 φ E = E φ E 2m x 2 2 k 2 2mE φ(x; E) = A si(kx) + B cos(kx). ψ(0) = ψ() = 0 k = π( + 1), so π k = ( + 1). This meas that the allowed eergies are π 2 2 ( + 1) 2 E =. 2m 2 Note that the eergies are discrete, ad that E 0 > 0, which is also thaks to the ucertaity priciple. This is very dieret rom a classical particle i a box! Also, the eigeuctios are φ (x) = A si(k x), ad the time-evolved eigeuctios are ψ (x, t) = A e iet si(k x). Note agai that or eergy eigestates, the probability desity p (x, t) = ψ (x, t) 2 = A 2 si 2 (k x) is idepedet o time. Also ote the ormalizatio requiremet meas that the coeiciets are ψ (x) 2 dx = 1 2 A = e iϕ.

5 8.04: Lecture 6 5 Figure 1: First, secod, ad third lowest-eergy eigeuctios (red) ad associated probability desities (blue) or the iiite square well potetial Oce agai, the overall phase is ot physical, so or coveiece, ϕ = 0, so that 2 A =. Ay good waveuctio ψ(x) at a give time t ca be expaded i terms o the eergy eigeuctios φ as ψ(x) = c φ(x; ) (0.2) or some c, where we ormalize (φ i φ j ) = φ (x; i)φ(x; j) dx = δ ij (0.3) so that the ormalizatio (ψ ψ) = c i c j δ ij = c j 2 = 1 (0.4) holds or the waveuctio. Goig back to the example o the iiite square well, the eigeuctios are 2 φ(x; ) = si(k x) i,j j

6 6 8.04: Lecture 6 with π k = ( + 1), so the expasio 2 ψ(x) = c si(k x) is just a usual Fourier series! Similarly, or a ree particle, the eigeuctios are with the ormalizatio 1 φ(x; k) = 2π e ikx (φ k φ k ' ) = δ(k k ' ), so the expasio 1 ψ(x) = ψ(k)e 2π is just a ormal iverse Fourier trasorm! Note that c = (φ ψ) is cosistet with ay geeral expasio ψ(x) = c φ(x; ), ikx dk ad the cotiuous aalogue is likewise true. We ca check this i the discrete case as c = (φ ψ) = φ (x; )ψ(x) dx = c s φ (x; )φ(x; s) dx s = c s δ s = c s as expected by orthoomality. We ca also check a example o the cotiuous case through the ree particle: ψ(k) = (φ k ψ) = φ (x; k)ψ(x) dx 1 ikx 1 ikx 1 ψ(x) dx = ik = e e ψ(k ' )e ' x dk ' dx 2π 2π 2π = ψ (k ' )δ(k k ' ) dk ' = ψ (k)

7 8.04: Lecture 6 7 agai by orthoomality. All operators correspodig to measurable observables have real eigevalues which are the values o those observables that ca be measured, ad the eigeuctios are orthoormal. Ay waveuctio represetig a quatum state ca the be expaded i terms o those eigeuctios: ψ(x) = c A φ(x; A) i the discrete case, or ψ(x) = A c(a)φ(x; A) da ˆ i the cotiuous case or a observable A with a correspodig operator A. The expasio coeiciets do have meaig. I the measurable values o a certai observable are discrete, the P(A) = c A 2 is the probability o measurig the value A or that observable, while i the cotiuous case, p(a) = c(a) 2 is likewise the probability desity o measurig that value. I the case o eergy, where the eigestates are give by ψ(x) = c φ(x; ), Êφ(x; ) = E φ(x; ). The probability o measurig the eergy to be E is thereore P(E ) = c 2. I the case o mometum, the expasio coeiciet is the Fourier trasorm o the waveuctio, so the probability desity o measurig a mometum p = k i that state is p(k) = ψ (k) 2. I the case o positio, the expasio coeiciet is exactly the same waveuctio at a dieretly-labeled positio because the positio eigeuctios are Dirac delta uctios, so the probability desity o measurig a positio x 0 i that state is p(x 0 ) = ψ(x 0 ) 2.

8 8 8.04: Lecture 6 But there is a eve better reaso to expad waveuctios i terms o eergy eigeuctios. I ψ(x, 0) = c φ(x; ), the ψ(x, t) = c e iωt φ(x; ) (0.5) describes how the state evolves i time. The reaso this works is because the eergy operator Ê ad the Schrödiger equatio respect superpositio! Note that i φ(x; ) is postulated to ot evolve i time, the the expasio coeiciets must evolve i time: iω c t (t) = c e. However, P(E, t) = c (t) 2 = c 2 is idepedet o time as the time evolutio is simply a complex phase. Similarly, (E) = E P(E, t) = E c 2 is idepedet o time. However, it ca be show that quatities such as (x) i geeral do deped o time. This is because physical state waveuctios are ot pure eergy eigeuctios but are superpositios thereo!

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