V (x) Box potential + bump. bump from - x to x Hight of bump V o

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1 Problems: 5. (c) nd (d); 5.3 ; 5. (modified see below) ; 5.5 (see hints in problem 3 below) ; 5.30 (see hints in 4 below) ; (5.7 + problem five below) ; Prctice Test # (Grded) () Replce (c) with the following: Clculte how the clssicl turning points (the plce where the prticle would bounce off the well wlls nd turn bck) depend on energy. Mke shure your grphs in prts () nd (b) reflect this dependence. (b) Replce (d) with estimte the size of the ground stte wve function for n electon in this potentil. Estimte the kinetic nd potentil energies ssocited with this wve function. Your nswer should involve C, nd m e.. For n even potentil, V ( x) V (x) (tke hrmonic oscilltor for exmple V (x) 1 kx ) rgue tht the wve functions must either be even or odd, i.e. Ψ n ( x) ±Ψ n (x). (1) Hint for symmetric potentil there is no preference to be either to the right or the left sy these words gin nd sk wht tht mens. Sketch the lowest four wve functions of the hrmonic oscilltor oscilltor verify tht the wve functions hve this property is the cse. 3. (Grded bsed on problem 5.6 in book). Consider n electron in box with smll dditionl potentil shown below: V (x) Box potentil + bump bump from - x to x Hight of bump V o / x / Here the width of the potentil x is much-much smller thn the size of the box x nd the potentil energy V o is smll. A good student sks: Wht mens smll? Smll mens tht the V o should be smll compred to something? Wht is tht something? Well, I will tell you. Smll in this cse mens tht V o should be smll compred to the typicl energy /M. When such smll potentil is dded, the energies, which re orignlly E n π n /M, chnge by reltively smll mount of V o. In fct, the chnge in energy is smller thn V o, becuse the width of the potentil is smll compred to the size of the box. Clerly, in the limit tht the potentil hs no width x 0 the originl energies re unmodified. This is the ultimte pproximtion. Thus we expect for smll x the energies to chnge by n mount of order V o ( x )somepower. We will show shortly in clss tht in the presence of smll dditionl potentil potentil δv (x) the energy of the n-th stte is chnged by the verge of the dditionl potentil. More specificilly, δe n δv (x). E n E n + δe n δe n δv (x) where ψ n (x) re the unpertubed wve functions, i.e. Eq. (9) ψ n(x) δv (x) ψ n (x) () 1

2 (hint on 5.5) When sketching the wve function, keep in mind how E V influences the bending of the wve function. () For the unperturbed wvefunctions of the prticle in the box, show tht close to the center of the box the probbility density is pproximtely { [ 1 (nπ) ( ) ] x n 1, 3, 5... P (x) ( ) (nπ) x (3) n, 4, 6... The tylor series of sin nd cos should be useful here. (b) Show tht the shift in energy is δe n { 4Vo x ] [1 (nπ) ( x) n 1, 3, 5... ) 3 n, 4, V ( o 3 (nπ) x Drw two energy level digrms side by side. The first showing the unperturbed energy levels. The second showing the perturbed energy levels. (c) Explin why the sptilly odd wve functions (to wit n, 4, 6...) experience less shift then the sptilly even wve functions (to wit n 1, 3, 5...) Wht would hppen if the perturbing potentil ws plced t the end of the box. Drw schemtic level digrm in this cse. (d) Explin why the energy shift for the sptilly even wve functions decrese with incresing n, while the energy shift for the sptilly odd wve functions increses with n. 4. (hints on 5.30) () In prt () think crefully bout the concvity of the wve function in different regions. (b) For prt(b) remember tht the wve will be loclized where the potentil is lowest. Remember tht since the potentils re even, the wve functions re even or odd. You might think crefully bout Problem 3 to decide wht configurtion would hve the lowest energy. This lst hint pplies to (d) nd (e). 5. Using the probbility density corresponding to the wve function in problem 7, determine the distnce D over which Ψ Ψ is smller thn its vlue t the edge by fctor of 1/e. Without looking up numbers, evlute D in ngstroms when the the electron is energy 1 ev under the potentil, i.e. E V 1eV. (4)

3 Wvefunctions 1. The electron wve function squred Ψ(x, t) P(x, t) is probbility per unit length to find the prticle t time t. Thus the probbility dp to find prticle between x nd x + dx t time t dp P(x, t)dx Ψ(x, t) dx (5). The electron must be somewhere so dx Ψ(x, t) 1 (6) 3. The verge position t time t x dx x Ψ(x, t) (7) 4. The verge position squred t time t is x dx x Ψ(x, t) (8) 5. The uncertinty squred in position ( x) (or stndrd devition squred) is defined to be ( x) x x (x x) (9) If the verge position is zero x 0 then ( x) x is the root men squre position. This gives mesure of how spred out is the wve function Momentum Averges 1. We use nottion for Opertors x dx Ψ (x)xψ(x) (10) dx Ψ (x)xψ(x) (11) Here X is n simply n opertor which tkes the function Ψ(x) nd spits out the new function xψ(x). It just gives nottion to things tht we lredy understnd, for exmple X Ψ(x) X xψ(x) x Ψ(x). The verge momentum is Here the momentum opertor is p dx Ψ (x)pψ(x) (1) ( dx Ψ (x) i d dx ( dx Ψ (x) i dψ dx P i d dx tkes the function Ψ(x) nd spits out the derivtive i dψ dx. ) Ψ(x) (13) ) (14) 3

4 3. The verge momentum squred is p Ψ (x)p Ψ(x) (15) ) Ψ (x) ( d dx Ψ(x) (16) 4. The uncertinty squred in momentum (or stndrd devition squred) is defined like for ( x) ( p) p p (17) Agin if p is zero then p p is the root men squre momentum. 5. The verge kinetic energy is KE Ψ (x) [ m d dx ] Ψ(x) (18) 6. The forml sttement of the uncertinty principle is ( x)( p) (19) where the stndrd devition in position x nd momentum p re defined s bove. (You cn see why its good thing tht we know how to use it before we cn stte it precisely) Quntum Mechnics 1. As with momentum, the verge energy of prticle is E Ψ (x, t)eψ(x, t) (0) [ dxψ (x, t) +i ] Ψ(x, t) (1) t nd the energy opertor is E +i t () Note the difference in sign between this nd the momentum opertor P which ultimtely is reflection of the fct tht wves re written e +i(kx ωt) with opposite signs for k nd ω. The Schrödinger eqution cn be written [ P [ M where V(x) is the potentil energy (think V 1/ kx ) 3. The sttionry wve functions (or eigenfunctions) hve the following form ] M + V (X) Ψ(x, t) EΨ(x, t) (3) d ] dx + V (x) Ψ(x, t) +i Ψ (4) t Ent i Ψ(x, t) e Ψn (x) (5) And re clled sttionry becuse the squre does not depend on time Ψ(x, t) Ent i e Ψ n (x) (6) Ψ n (x) (7) Since the wve funcion does not depend on time we cn tke long time to determine the energy. So sttiony hve functions exctly the energy E E n. There is no uncertinty in the energy. E E E 4

5 4. Sttionry wve functions (lso known s eigenfunctions) obey the time independent Schrödinger eqution. [ d ] M dx + V (x) Ψ n (x) E n Ψ n (x) (8) Here the E n re re the energy levels (s in the E n 13.6eV/n in the Bohr Model) re the E n in the sttionry wve functions Prticle in the Box 1. For n electron bouncing round in box of size the sttionry wve functions (eigen-functions) re Ψ n (x) cos ( ) nπx n 1, 3, 5,... sin ( ) nπx n, 4, 6,... while the sttionry energies re Qulittive Fetures of Schrödinger Eqution (9) E n k n M π M n n 1,, 3, 4, 5,... (30) [ m d ] dx + V (x) Ψ n (x) E n Ψ n (x) (31) 1. In the clssiclly llowed region the egein-functions oscilltes. In the clssiclly forbidden region the wve function decys exponentilly. For given potentil you should be ble to roughly sketch the wve functions.. In the clssiclly llowed region E > V, you be ble to show (by ssuming tht the potentil V is constnt) tht the wve function oscilltes with wve number k π/λ s m(e V ) Ψ(k) A cos(kx) + B sin(kx) where k (3) 3. In the clssiclly forbidden region E < V, we sy tht the prticle is under the brrier becuse E < V. Assuming tht the potentil is constnt you should be ble to show wve function decreses s m(v E) Ψ Ce κx where κ, (33) s one goes deeper into the clssiclly forbidden region. The length D 1/(κ) is known s the penetrtion depth. It is the length over which the probbility decreses by fctor 1/e. 4. Looking t the Schrödinger eqution we conclude: d ψ dx m (E V )ψ (34) () In the clssiclly llowed region (E V > 0) the wve function is concve down if ψ > 0 nd concve up if ψ < 0. The strength of the curvture (how rpidly it oscilltes) is controlled by m(e V )/. (b) In the clssiclly forbidden region (E V < 0) the wve function is concve up if ψ > 0 nd concve down if ψ < 0. The strength of the curvture is controlled by m(v E)/. 5. The energy is determined in the Schrödinger Eqution by demnding tht the wve function Ψ decreses s x. This is why we neglect exponentilly incresing solutions e +κx in the preceding item. 6. Generlly there is one more hlf wvelength in the box (or more properly the clssiclly llowed region) ech time the prticle in the potentil is excited from one energy stte (i.e. Ψ n ) to the next higher one (i.e. Ψ n+1 ). Look t Box wve functions or simple hrmonic oscilltor wve functions for exmples. 5

6 7. The wve function is lwys continuous nd hs continuous first derivtives. 8. An exmple of these trends is given by the set of simple hrmonic oscilltor wve functions hnded out in clss. You should be ble to see these fetures in the figures hnded out in clss. Perturbtions 1. In the presence of smll dditionl potentil potentil δv (x) the energy of the n-th stte is chnged by the verge of the dditionl potentil. More specificilly, δe n δv (x), i.e. E n E n + δe n δv (x) where ψ n (x) is the unperturbed wve functions. ψ n(x) δv (x) ψ n (x) (35). The wve functions re orthogonl. Orthogonl mens in the context of quntum mechnics: { ψm(x)ψ 0 n m n (x) 1 n m (36) 6