Shallow Square Well I
المؤلف:
Sidney B. Cahn, Gerald D. Mahan And Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 2 , p 51
21-8-2016
1530
Shallow Square Well I
A particle of mass m moving in one dimension has a potential V(x) which is a shallow square well near the origin:
(1)
where V0 is a positive constant. Derive the eigenvalue equation for the state of lowest energy, which is a bound state (see Figure 1.1).
Figure 1.1
SOLUTION
The ground state energy E must be less than zero and greater than the bottom of the well, 0 > E > -V0. From the expression
(1)
one can deduce the form for the eigenfunction. Denote the ground state energy E = -h2α2/2m, where α is to be determined. The eigenfunction outside the well (V = 0) has the form exp(-α|x|). Inside the well, define k2 = k20 – α2, where k20 = 2mV0/h2. One can show that k2 is positive since E + V0 > 0. Inside the well, the eigenfunction has the form A cos kx, so
(2)
Matching ѱ(x) and its derivative at x = a gives two expressions:
(3)
(4)
Dividing these two equations produces the eigenvalue equation
(5)
The equation given by the rightmost equals sign is an equation for the unknown k. Solving it gives the eigenvalue E.
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