Schrödinger, 3. 2 y + 2. π y / L 2. )sin(n y. )sin(n z. n x. = 2 π 2 2 L 3 L 2 / 8.

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1 Schödinge, 3 The 3D infinite squae well: quantum dots, wells, and wies In the peceding discussion of the Schödinge Equation the paticle of inteest was assumed to be moving in the x -diection. Of couse, it is not possible fo a paticle to be moving in one spatial diection only. If that wee tue, accoding to the HUP it could be anywhee in the y - and z -diections and theefoe be undetectable with finite volume detectos. Now, we conside the moe ealistic case of motion in all thee spatial diections. Fo this pupose, we stat with the 3D infinite squae well. This model povides a useful conceptual famewok fo undestanding a bugeoning nano-industy, namely the poduction and use of quantum dots. The 3D infinite squae well is a igid ectangula solid box. The potential enegy of a paticle tapped inside is given by U(x, y,z) = 0, if 0 < x < L 1, 0 < y < L, 0 < z < L 3, and, othewise. The paticle s momentum now has thee components and its kinetic enegy is K = ( p x + p y + p z ) m. Afte eplacing each momentum component with an appopiate diffeential opeato, the wavefunction satisfies the 3D Schödinge Equation i Ψ = t m x + y + z Ψ (1) inside the well and its eigenstates can be sepaated into Ψ(x, y,z,t) = X(x)Y (y)z(z)t (t), whee, as befoe, T (t) exp( iet / ), and X,Y, and Z ae sine functions that vanish on the walls of the well: Ψ nx n y n z (x, y,z,t) = Asin(n x π x / L 1 )sin(n y π y / L )sin(n z π z / L 3 )exp( ie nx n y n z t / ). In this expession, all the n s ae positive integes (i.e., 1,, 3, ) and the enegy eigenvalues ae E nx n y n z = π n x m L + n y 1 L + n z L 3. () The nomalization constant A is detemined by equiing ae ove all possible values of x, y, and z : A = L 1 L L 3 / 8. Ψ dxdydz = 1, whee the integals Example: Suppose the tapped paticle is an electon and the dot is cubical with L = 1 nm. What is the electon s gound state enegy? Solution: The gound state has the lowest enegy, coesponding to all of the n s being equal to 1. Thus, E 111 = 3 c ( ) π mc L, about 1 ev. By choosing dot sizes coectly, one can ceate dots that absob o emit photons of welldefined enegies. The image to the ight (best viewed in colo) shows a dozen vials contained CdSe quantum dots of diffeent sizes in solution. Afte absobing UV adiation these dots emit in the colos shown. By coating the CdSe dots with poteins that attach to specific cell membane molecules it is possible to tace out cell stuctues by illuminating the cells in white light and looking at the emission at the appopiate wavelengths. This technique has been shown to be effective in Sc3 1

2 mapping tumos in test animals (Natue Biotechnology,, (004)) and holds geat pomise fo clinical application fo humans. Also see, Dazzling Dots, Science News, pp -5, July 11, 015. Note that the states of a paticle in a 3D well ae associated with thee quantum numbes one fo each way the paticle can move in space. A shot table of lowest enegy eigenvalues fo a cubical well is n x n y n z E n x n y n z E In the table E is measued in units of π ml. In the table, E has a single enty fo two states, (111) and (), othewise all othe enties have 3 o 6 diffeent states with the same E. This situation is called degeneacy. An enegy level that has N possible states is said to be N -fold degeneate. Thus, the enegy levels in the table with two 1s and a ae 3-fold degeneate; the levels with all possible combinations of 1,, and 3 ae 6-fold degeneate. Levels that have only one possible state ae nondegeneate. Degeneacy aises fom two causes: symmety and an aithmetic accident. In all of the cases in the table, the degeneacy aises fom the fact that fo a cube you can switch the labels of the x, y, and z -axes without changing anything. But that s not tue if the sides of the ectangula solid ae unequal. If, instead of L 1 = L = L 3 = L as in the table, we have L 1 = L, L = L /, L 3 = L / 3 (so that the nice cubical symmety is emoved) then only the states (11) and (1) in the table ae degeneate (with E = 15), and that is meely a numeical accident having nothing to do with symmety. A 3D well becomes effectively a D well if one of the sides is much smalle than the othe two. Fo example, suppose L 1 = L = L, L 3 = L /10. The enegy levels would then be, E = E 0 (n x + n y +100n z ), whee E 0 = π ml. Thus thee would be lots (a few hunded) of combinations of diffeent n x and n y values with n z = 1 that would have lowe enegies than fo (11), the lowest enegy state with the fist level of excitation in the z -diection. At low enegies, the z motion is effectively fozen, and the well acts as if it wee D. That s a so-called quantum well. A simila thing happens if two of the sides ae much smalle than the thid. Then at low enegy two of the motions ae fozen and only the thid has the possibility of excitation. In that case, that s a 1D quantum wie. Sc3

3 The sanitized hydogen atom Though they povide useful qualitative undestanding, the squae well potential enegies we have studied ae simple appoximations to the eal inteactions between paticles. We now apply what we have leaned about the quantum behavio of paticles with squae well potential enegies to undestanding the atomic stuctue of matte. Atoms ae bound systems of negatively chaged electons and positively chaged nuclei. The most elementay kind of atom is hydogen consisting of one electon (with electic chage e, whee e = 1.6x10 19 C) and one poton (with electic chage +e ). The simplest vesion of the hydogen atom assumes the electon inteacts with a point-like poton sitting motionless at the oigin of some coodinate system via the attactive Coulomb (electostatic) potential enegy U ( ) = k E e, (3) whee is the instantaneous distance between the electon and poton and k E = 1 4πε 0 is the electostatic foce constant. This simple pictue ignoes the facts that the electon s motion has elativistic coections, the poton has a finite size (indeed, consists of smalle paticles called quaks and gluons) and actually moves as the electon moves (in fact, it can t be at est at an exact position because that violates the uncetainty pinciple), and that the poton and electon inteact magnetically, though the weak foce, and though gavity (which, compaed with the Coulomb inteaction, ae small, eally small, and eally, eally small, espectively). All of these ignoed aspects poduce only mino modifications to the geat tiumph of the Schödinge desciption of hydogen, namely, evaluation of the electon s enegy and angula momentum eigenvalues. (Enegy eigenvalues ae associated with the discete wavelengths of photons emitted when excited hydogen atoms de-excite and angula momenta ae measued in expeiments whee hydogen atoms ae exposed to stong extenal magnetic fields. Pehaps you have studied a catoon model of the hydogen atom known as the Boh atom. In it, the electon obits the poton in cetain allowed classical cicula obits that ae kludged in just the ight way to explain hydogen s emission specta. Unfotunately, this catoon model gets hydogen s obseved angula momenta completely wong. The Boh atom was invented befoe Schödinge quantum mechanics and, though it pedicts quantized enegy values, is actually incompatible with it. Fo example, a classical cicula obit has a pecisely defined adius. But the Heisenbeg Uncetainty Pinciple dictates that zeo uncetainty in adius equies total uncetainty in adial momentum and, theefoe, obital enegy. In quantum mechanics it is not possible to know both simultaneously.) As (a) total mechanical enegy equals kinetic plus potential enegies and (b) the electon in the hydogen atom moves in thee dimensions, it seems plausible that the Schödinge Equation fo the hydogen poblem is just equation (1) above with the potential enegy in equation (3) added to the ight hand side. But thee s a mathematical complication. In Catesian coodinates, = x + y + z. Thus, thee ae infinitely many diffeent x, y, and z values that can be combined to give the same. The Coulomb potential enegy depends only on the length of the electon s position vecto, not on its diection; it is spheically symmetic. This potential enegy mixes x, y, and z in a complicated way. As a esult, the Schödinge Equation fo the Coulomb potential enegy cannot be solved by sepaation of vaiables if Catesian coodinates ae used. In geneal, wheneve the potential enegy is spheically symmetic one has to use spheical coodinates:,θ,φ. Sc3 3

4 Spheical and Catesian coodinates ae elated as shown in the figue to the ight. The coodinates of the tip of the position vecto ae eithe ( x, y,z) o (,θ,φ), whee the two ae elated by x = sinθ cosφ y = sinθ sinφ z = cosθ A classical paticle moving in 3D has a momentum p that, in spheical coodinate language, is patly adial (along a line passing though the oigin of coodinates, oncente ) and patly tangential ( off-cente ): p = p + p tan, whee p and p tan ae pependicula to each othe (see ight). The classical kinetic enegy of the paticle is given by K = p m = p p m = p m + p tan m = K + K tan, that is, a pat due to adial motion and a pat due to off-cente motion (i.e., aound the oigin). The latte is associated with angula momentum. Angula momentum is defined as L = p. It is a measue of motion about the point that emeges fom. Because the coss poduct of two vectos that lie along the same line is zeo p = 0 and, hence, L = p tan. The magnitude of L is the poduct of the magnitudes of and p tan (because the angle between them is. natually 90 ): L = p tan. Fo a cicula obit L, calculated about the cente of the cicle, is pependicula to the plane of the obit and points along you ight thumb if you cul you ight hand finges in the diection of the motion [see ight]. In geneal, d L dt = d ( p ) = d dt dt p + d p. The fist tem on the ightmost side is zeo dt because d dt = p m. The second tem is the toque: τ = F net. In the hydogen poblem, the Coulomb foce of the poton on the electon points diectly down the adial diection, so d L dt = 0. As the electon obits the poton its obital angula momentum is a constant (both in diection and magnitude). We can eplace p tan by L in the paticle kinetic enegy to obtain: K = p m + L m. When that s done, the Schödinge Equation fo the sanitized hydogen poblem is fomally E op Ψ = K op Ψ +UΨ = K,op Ψ + K ob,op Ψ +UΨ = K,op Ψ + L op Ψ m k E e Ψ. (4) Afte eplacing (that is, afte doing a lot of tedious algeba) the deivatives with espect to x, y,z ( ) in the Schödinge Equation by deivatives with espect to (,θ,φ) (e.g., using x = x + θ x θ + φ, and so on) we obtain x φ i Ψ = 1 t m Ψ 1 Ψ + sinθ sinθ θ θ + Identifying the opeatos in (4) with the deivatives in (5) yields: 1 sin θ Ψ φ k E e Ψ. (5) x O φ θ p z p y p tan Sc3 4

5 E op = i t K,op = 1 m 1 L op = sinθ θ sinθ 1 θ + sin θ φ Sepaating the vaiables in (5) via the poduct Ψ(,θ,φ,t) = R( )Θ( θ )Φ( φ)t t secets of the (sanitized) hydogen atom. ( ) eveals the Sc3 5