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Unformatted text preview: : ér/y/éﬂg) “(/Zﬁéﬂéc ,7; Mat/{£41
Wee/QM“ J} ) 9336/ (M F INITE _S QUAKE WELL As a last example, consider the ﬁnite square well potential  ._ —V0, for —a<x<a,
‘V(x) _ { 0, for x > _a, [2'145] where V0 is a (positive) constant (Figure 2.17). Like the deltafunction tat/ell, this
potential admits both bound states (with E < 0) and scattering states (with E > O). We'll'look ﬁrst at the bound states. ..
In the region x < —a the potential is zero,'so the Schrodinger equation reads it: dzw H 7
ﬂ"— : a = u a
2171 d):2 w or tt'JL'2 K Where; ' .
«/—2 E
K E f’“ [2.146]
1 is real and positive. The general solution is (bot) = A exp(—Kx) + B expch), but
the ﬁrst term blows up (as x —>’ —oo), so the physically admissible solution. (as before_—_see Equation 2.119) is (bot) = Be“, for x _< —ct. [2147] ‘  ' FIGURE 2.17: The ﬁnite square well
(Equation 2.145). ' In the region —a < x < a, V(.r) = —V0, and the Schrodinger equation reads hi dzvx div " 7
—— — V = E , = l ,
2m (Lt2 0w 1’0 or dx2 1/}
where
' . ﬂ/2m(15 + V )
ls [2.148] Although E is negative, for bound states, it must be greater than —Vﬁ, by the . old theorem E > Vmin (Problem 2.2); so I is also real and positive. The general solution is” doc) = sinUx) + D (20503:), for — a < x a, ' [2.149] where C and D are arbitrary constants. 'Finally, in the region .1: > a the potential
is again zero; the general solution is iﬂx) = F e'xp(—icx) + Gexpch), but the
second term blows up (as x ——> 00), so we are left with wot) '= Fez—"'1', for x > a. [23150] I he next step is to impose boundary conditions: w and div/(ix continuous at
—a and +n. But we can save a little time by noting that this potential is an even
function, so we can assume with no loss of generality that the solutions are either
even or odd (Problem 2.1(c)). The advantage of this is that we need only impose the boundary conditions on one side (say, at +a); the other side is then automatic,
since w—x) = i‘iﬁx). I’ll work out‘the even solutions; you get to do the odd
ones in Problem 229. The cosine is even (and the sine is odd), so I’m looking for solutions of the form Fe'“, for x > a,
1,!!(x) = Dcosﬂx), for O < x < a. [2.151]
iM—x), for .r < 0. ‘ The continuity of wot), at x = a, says Fe'm = D cos(!a), [2.152] and the continuity of dw/dx, says _‘
—icl"e'm = —ID sin(la).  [2.153]' Dividing Equation 2.153 by Equation 2.152, we ﬁnd that '
I  x = l tanUa). . [2.154] it}? it 3M2 2n ' 57cm 2 '5 FIGURE 2.18: Graphical solution to Equation 2.156, for zo = 8 (even states). This is a formula for the allowed energies, since K and l areboth functions
of E. To solve for B, we ﬁrst adopt some nicer notation: Let 2: E In, and zo :—: ;,/2ml/o. [2.1551
1
According to Equations 2.146 and 2.148, 051+ 13) = 2m Vo/hz, so are = 1% — zz, and Equation 2.154 reads tanz: 1! (zo/z)2 — 1. _ [2.156] This is a transcendental equation for z (and hence for E) as a function of zg
(which is a measure of the “size” of the well). It can be solved numerically. using a computer, or graphically, by plotting tanz and ./(zo/z)2 — 1 on the same grid,
and looking for points of intersection (see Figure 2.18). Two limiting cases are of special interest:
1. Wide, deep well. If 2:0 is very large, the intersections occur just slightly below :5" = tut/2, with it odd; it follows that “En.th ' ‘ '
———— I [2.157] I E V 2 .
"+ 0 2m(2ct)2 But E + V0 is the energy above the bottom of the well, and on the right side
we have precisely the inﬁnite square well energies, for a well of width 20 (see
Equation 2.27)—or rather, half of them, since this It is odd. (The other ones, of ‘
course, come from the odd wave functions, as you’ll discover in Problem 2.29.) So
the ﬁnite square well goes over to the inﬁnite square well, as VG —~> .00; however.
for any ﬁnite V0 there are only a ﬁnite number of bound states. 2. Shallow, narrow well. As 20 decreases, there are fewer and fewer bound
states, until ﬁnally (for zo < Ir/Z, where the lowest odd state disappears) only one
remains. It is interesting to note, however, that there is always one bound State, no matter how “weak” the well becomes; ...
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 Spring '11
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