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i.e. AB¢ == BA'i' and [A, B] == O. (b) Let ip be the eigenfunction of the operator Ii and belong to the eigenvalue A. Since A and iJ are commutative, AB'¢ === BA1p == = BA'lf' = AB,¢, i.e, A'¢' == A'¢', where 'P' === B~J. Thus, the eigen- value A belongs both to the function '¢ and to the function '¢' that describe therefore the same state. This can happen only when these functions differ by a constant factor, for example, B: 'P' = B'¢. But 'P' = B'1', and therefore B'¢ === B,¢, i.e. the function '¢ is the com- mon eigenfunction of the operators A and iJ. 4.14. (8) f (x, z) eikyY; (b) Aei(kxx+kyY+kzz}; (c) f (Y, z) e±ikxx. Here k i === Pi1h (i === x, y, z); t is an arbitrary function. 4.15. It does only when the function '¢A is at the same time the eigenfunction of the operator iJ. I t does not in the general case. For example, in the case of degeneracy (in a unidimensional rectangular potential well two values of the momentum's projection, +p~ and -Px, correspond to each energy level despite the fact that the op- A "",l erators Hand P:x commute). 4.16. Suppose that '1-' is ap. arbitrary eigenfunction of the oper- ator A, corresponding to the eigenvalue A. Then from hermi- city of the operator A it follows that J'I\J*A'I\J dx = J'l\JA*'I\J* dx and A J'I\J*'I\J dx = A* .\ 'I\J'I\J* dx, whence A = A*. The latter is pos- sible only if A is real. 4.17. <a) J'l\JtPx'I\J2 dx = -in j 'I\J~ aa~2 dx = - in [('I\J~'l\J2r!:: - - ~ 'l\J2 a:; dx ] = J'l\J2 (in :x) 'I\J~ dx = J'l\J2pN~ dx, 4.18. The operator A+ conjugated with the operator A is de- fined as follows: J'I\J:A'l\J2 dx = J'l\J2 (A+'l\Jt)*dx, (a) Pxx; (b) -iPx. 4.19. From the hermicity of the operators Aand iJ it follows that J'I\J:A(B'l\J2) dT = JB'l\J2 (A*'I\J:) dT = JA*'I\J: (B~2) dT = J'l\J20* (A*'I\J*) dx . Since A and B commute, B* AA* = A*B* and J'I\J:AB'l\J2 dT = J'l\J2A*B*'I\J: dx, 167 4.20. Every operator commutes with itself. Consequently, if the operator.it is hermitian, the operators A2 = AA and An are also hermitian. 4.23. (a) The solution of the equation Li¢ = Lz'P is 'P (CP) = = AeiLzQ)/n. From the condition of single-valuedness 'P (CP) = = 'P (cp + 2n) it follows that L z == mh, where m = 0, ±1, ±2, .... From normalization, we get A == (2n)-1/2. Finally, 'Pm (cp) = = (2n)-1/2ei m q> . (b) The eigenvalues are L; == m21i2 , where m = 0, +1, ±2, .... The eigenfunctions have the same form as for the operator t. z, i.e, the function 'Pm (cp) = (2n)-1/2ei m cp is the common[eigenfunction of the operators Lzand i; All states with eigenvalues L~ except for m = 0 are doubly degenerate (in terms of direction of the rotational moment, L z = +1min). 4.24. 21i2• 4.25. (a) J'iJ:Lz'iJ2dq> = - in ('iJ:'iJ2)~3t + J'1'2 ( in a: )dq> = = J'iJ2i Nt dq>. Here ('¢t'¢2)gn = 0, because the functions '1'1 and '1'2 satisfy the condi- tion of single-valuedness. (b) J'iJ~iz'l'2d't= J('iJ:XPy'l'2-'iJ:YPx'l'2) dx, Since the operators Px and Pu are hermitian, the integrand can he transformed as X'P2P~'P~- Y'P2P~'!': = ~2 (xp~ - yp~) ~': == 'P2L~'i':. - 4.26. J,~~b'hd't=" J('iJ:iN2+'iJ:LN2+'iJiLN2)d't. Since the operators L; I:y , and L, are hermitian, the squares of these op- erators and consequently the operator £2 are also hermitian. 4.29. (8) [Lx, pi] = [Lx, p::l pAX + p: [ix, Px] = 0, for [Lx, Px]=O. 4.30. The operator T can be represented in the spherical coor- dinates as the sum f = i r +L2/2mr2, where TAr performs the op- eration only on the variable r, Since the operator £2 == - 1i2V~, q> operates only on the variables '6' and fl), [£2, T] == [i 2 , Tr] + + [£2, £2/2mr2 ] == o. 4.31. (8) [ix, i y) == LxLy- LyLx== (yp1,- zpy) (zPx- xpz)- - (zPx- xpz) (ypz - Zpy) == [z, Pzl (xp y - YPx) = in (xPu - YPx)= = iliL z• 4.32. (a) [i2 , Lx] ~ [Li, Lx] + [i~, Lx] + [i;, Lx], where [i~, Lxl = 0; [i~, Lxl = Lg [Lg , Lxl+ [iy,·Lxl L, = -in (Lyiz + 168 +LzLy); [i~, Lx] = i., liz, Lx] + [L z, l:x] LAZ == in (izLy+ iyLt ) " Whence it follows that [£2, Lxl == O. Similarly for L, and Lz. 4.33. In the case of r == r o==const, the operator ,. 1i2 1 A H= --22 V~,CP==-22 L2..J-lro ~ro Therefore, iI'lh = _1_ L2'lh = E1h. 't' 2~r~ 't' 't' Since the eigenvalues of the operator £2 are equal to 1i2 l (l + 1)~ then E === 1i2 l (l + 1)/2J.tr~. 4.34. (a) Since the operator A is hermitian, J'IjJ*A'IjJ d't = = I 'ljJA*ljJ* dx, Consequently, (A) = (A*), which is possible only for a real (A). 4.36. Hx - xiI = - ~ Px, therefore (Px) = r 'i'*Px'l' dx == m J = - ~ J('IjJ*HxljJ- ",*xH'IjJ) dx. Due to hermicity of the Hamil- • A A tonian the integrand can be written as x'¢H'¢* -x'¢*H'¢ == 0, for iI,¢* == E1P* and H'l' == E'¢. ~ 4.37. (8) From the normal iz ing conditions A2 == 8/3l. r A 1i 2 r d2'¢ 2 n 2 Ji 2(T)= J ~'\T'¢dx= - 2m J '¢ dx2 dX==3·~· (b) A2= 30/15 ; (T) == 51i2/m12• 4.38. From the normalizing conditions A2 == Y 2/n a; (T) =- = (U) = nw/4. 4.39. (8) Here 'i'n (x) = Y 21l sin (nnx/l), «dX)2) = (x2) - (X)2 = = :; (1 - n~n2) ; «L\px)2) = (pi) = (1tft/l)2 n2• (b) From the normalizing conditions A2= V2/n"a; «(LlX)2) == = 1/4a2 ; «(l\Px)2)= a 21i2• (c) A2 ==V' 2/1t a; «(LlX)2) == (x2) == 1/4a2 ; «(LlPx)2) == (p~) - (Px)2 = == a,21i,2. Instruction. While calculating the mean value of the momentum squared, it is helpful to use the following hermitian property of the operator Px: (pi) = J'IjJ*pi'IjJdx= J IPx'IjJ12dx. 4.40. From the normalizing conditions A 2 = 4/331; (L~) = 4h,2/3. 4.41. From the normalizing conditions A2 == 1/n; «(~<p)2) = = (<p2) _ (cp)2 == n 2/3 - 1/2; «(dL z)2 ) = (L~) == ti2• 169 4.42. Recalling that ini.; = [i y , izl, we can write (Lx> = 1 \ A A A A -= "7 ('¢*LyLz'P - '1'*LzLy~~) dx ; Since according to the condition l/~ c. Lz'¢ == Lz'l' and the operator i., is hermitian, the integrand can he transformed as Iol lows: '¢*tyLz1.p-~,*Lziy'¢==Lz'¢*.iy'¢ - (L u'1' ) i~'¢* := (L y '1' ) (Lz'i'* - L~'¢*). The latter expression in pa- rentheses is equal ~to zero because an eigenvalue of a hermitian operator is real (L z == L:). Similarly for the operator Ly • 4.43. (L2) = ) 'lj.lbljJ dQ = 21i2 , where dQ = sin 'I'} d1tdq>. 4.44. Since the axes x, y, z are equivalent, (L2) ==: (L~)+ (L~)+ + (L~) == 3 (L~). Allowing for equiprohability of various possible values of L z, we obtain 1. 2(L~) == Ii (m2) ==~-21+1 " m 2 == ,21i 2 L.J 21 + 1 m=-l '1 (l +1) (21 + 1) 6 and (L2) == Jj,2l (l + 1). 4.45. We have A'P1 == At 1.Pt and A'¢2 = A2'P2. Due to hermiclty of the operator A, its eigenvalues are real and ) ljJ~A'1'2 d-c :J 'lj.J2A*1\J: dx, or A21~):¢2 d-c = Ai ) 'lj.J2'Ij.J~ ds, Since At =1= A2 , the latter equality is possible only under the con- dition J1\J:1\J2 d-c= 0, i.e. the functions 'lj.Ji and 'lj.J2 are orthogonal. 4.47. (a) Multiply both sides of the expansion ¢ (x) = ,>; Ck'l'A (x) by 'Pc (x) and then integrate with respect to x: 1'lj.JtljJ dx = ~ Ck J ~)Nk dx. 'The eigenfunctions of the operator A are orthonormal, and therefore all integrals in the right-hand side of the equation, with the exception of the one for which k = 1, turn to zero. Thus, C j = 11\Jr (x) 1\J (x) dx, (b) {A} = 11\J*A'Ij.J dx = J(~ cN~) (~ CjA,'Ij.J,) dx = ~ c%cjA,X k,l X 1"'~'Ij.J, dx= ~ ICk /2Ak ' Note that ~ ICk I2= 1 which follows directly from the normalization of the function 1.1' (x): ) 'Ij.J*'Ij.J dx= ~ ctci J'lj.JNI dx= 1. 170 From here it follows that the coefficients J Cit 12 are the probabilities of observing definite values of the mechanical quantity A h- 4.48. First, the normalizing coefficient A should be calculated. The probability of the particle being on the nth level is defined by the squared modulus of the coefficient Cn of expansion of the function ¢ (x) in terms of eigenfunctions 'Pn (x) of the operator 8: en= = \ '¢ (x)'¢n (x) dx, where '¢n (x) = V2/l sin (nnx/l). (a) A 2 = 8/31. The