practice Midterm+1+Key

practice Midterm+1+Key - l The formula below can be derived...

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Unformatted text preview: l The formula below can be derived from equations we have discussed. 1 1 I “3,4,5, To which quantum mechanical problem does it apply? a) blackbody radiation b) photoelectric effect atomic hydrogen spectrum d) particle in a box e) harmonic oscillator 2. If (/11 and gyz are normalized cigenfilnctions of an operator [5 belonging to different eigenvalues fl and f2 of the operator, then the integral IVI, 1/12 dx: equals all > space 0 b) 1 c) 1/5 d) fl + f2 3. Which of the following is not a requirement for a valid wavefunction ‘PQQ? a) The wavefimction must be an eigen function of the Hamiltonian operator. b) For any value of X, LFog can have only one value. c) The wavefimction must be continuous. @ The energy of ‘i’(x) cannot be the same as that of any other valid wavefunction. 4. Foraparticleinabox,whereo<=x<=a,mstaten=3 ! = (if mm“) a) What is the probability that the particle will be located at X = a/3‘? b) Find <x> at) LVW’A) : [£)ml(3££> a, 3-4» 3 & $01217 W :0 a! as (0) MIL Ln Macy Q/ : fl 3flx jg “(3)304 [%>M 0” ma QJ a. : 7a.: 0 x4144 £1 fl ,, 0b )l/7. [ l 5. Show that W306) mfg?) is a solution to the Schrodinger equation and solve for E in terms of the other constants. (121/1 87rsz abc2 + [ 1.2 ] W 3 D mm : (5/; m (Lg) 3/17 (9/2 z w) (2%) “(a)”; My 191‘) W r 7— . L“) MAKE/X) + WW") : wan”) : 0‘“ 0v 6. The angular momentum operators are: I; z * mp f: a J x az "’zé—y“ ‘ _ _ 5 6 A . a a Showthat [i,,Ly]= m L Some Postulates and General Principles of Quantum Mechanics 93 ___________________._..———-————————-— 4—17. Referring to Table 4.1 for the operator expressions for angular momentum, show that [12“ fly] 2 mil try, L1] 1?th and [if 12x1=miy (Do you see a pattern here to help remember these commutation relations?) What do these expressions say about the ability to measure the components of angular momentum simultaneously? H __. 1-1 gay L*L"f’l ‘hlyaz Zaylll ‘hlzax "azll :_h2(y3f 6f x 32f 23—2f_ x 32f) ' f In the same way, we can show [11)” £1] = ihix and [1:3 11;] = [h if. The pattern involves the cyclic permutation of x, y, and 1. Since no combination of the operators ix, 1),, and £1 commutes, it is not possible to simultaneously measure any two of the three components of angular momentum to arbitrary precision (as discussed in Section 4—6). ...
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practice Midterm+1+Key - l The formula below can be derived...

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