6.90. a = mw/2h, E = hco/2, where co = 1/kInt.6.91. E = —me 4 18h (^2) , i.e. the level with principal quantum
number n = 2.
6.92. (a) The probability of the electron being at the interval
r, r + dr from the nucleus is dP = 2 (r)42-cr 2 dr. From the condi-
tion for the maximum of the function dP/dr we get rp ,. =
(b) (F) = 2e 2 /ri (^2) ; (c) (U) =
6.93. (p 0 = (p/r) 4nr 2 dr =-elri, where p =-4 2 is the space
charge density, ip is the normalized wave function.
6.94. (a) Let us write the solutions of the Schrodinger equation
to the left and to the right of the barrier in the following form:
x < 0, 14)i (x)=aieihiH- bie-ikix, where k (^1) = V- 2mElh,
x> 0, 2 (x) = a 2 eih2x b 2 e-ih2s, where k (^2) = -1(2m (E —U 0 )1h.
Let us assume that the incident wave has an amplitude a (^1) and the
reflected wave an amplitude b 2. Since in the region x >0 there is
only a travelling wave, b 2 = 0. The reflection coefficient R is the
ratio of the reflected stream of particles to the incident stream, or,
in other words, the ratio of the squares of amplitudes of correspond-
ing waves. Due to the continuity of* and its derivative at the point
= 0 we have a 1 b 1 = a 2 and al — bi = (k 2 1k 1 ) a 2 , whence
R =(b1/a1)^2 = (ki— k^2 )^2 /(k^1 + k2) 2.
(b) In the case of E < U 0 the solution of the SchrOdinger equa-
tion to the right of the barrier takes the form
11, (^2) (x) = a 2 e" b 2 e--", where x =2m (U 0 — E)1h.
From the finiteness of (x) it follows that a 2 = 0. The probability
of finding the particle under the barrier has the density P2 (x) =
114 (x)— e-2". Hence, xefi = 1/2x.
6.95. (a) D exp [— (Uo — E)];
(b) D exp[-^81 Ii2m 3uo (Uo —E) 312 ].
6.96. D exp [ — 21 -1/h 1 - 2-4-(' 7 - (U 0 —E)].
6.97. —0.41 for an S term°and —0.04 for a P term.
6.98. cc = jihR/(E 0 —eq)i)— 3= —0.88.
6.99. Eb=h111(1 1 Rki X, 2 122tc6,X— 1)^2 = 5.3 eV.
6.100. 0.82 μm (3S —)-2P) and 0.68 p,m (2P^ 2S).
6.101. AE = 21chcA2s,/k 2 = 2.0 meV.
6.102. Au) = 1.05.10" rad/s.
6.103. 3S 112 , 3P112, 3P (^312) , 3D312, 3 D512.
6.104. (a) 1, 2, 3, 4, 5; (b) 0, 1, 2, 3, 4, 5, 6; (c) 1/2, 3/2, 5/2,
7/2, 9/2.