Quantum Gravity 218.2: The Schrodinger Potential Step

From Osher Doctorow

Take a look at Wolfram's "Potential Step," where the potential V is:

1) V = 0 for x < 0
V = Vo for x > 0 (Vo > 0)

The potential V plotted against x yields the Schrodinger wavefunction
solution for x < 0:

2) w1 = A cos(kx) + B sin(kx)

For x < 0 we get the Schrodinger wavefunction solution:

3) w2 = Cexp(qx) + Dexp(-qx)

Note that the graph of V (vertical) vs x (horizontal) is the left ray
(excluding the origin (0, 0) V = 0, x < 0, that is to say the negative
x axis (since V = 0), while for x > 0 the graph is the half line V =
Vo parallel to and above the x axis and to the right of the V axis.

As one reader remarked in replying to a very recent posting of mine in
which I attacked trolls (I've forgotten his name or screen name - you
can look it up), we can use more information to solve for unknowns, as
for example here Wolfram uses the asymptotic requirement that w2
decays to 0 as x --> infinity and w1(0) = w2(0) and w1 ' (0) = w2
' (0) (derivatives at 0). That solves for A, B, C, D (constants) in
the above equations, and various other things can be assumed to
further reduce the unknowns in the equations (such as a traveling wave
incident to the right, producing reflected and transmitted waves).

Notice that the cosine, sine, and exponential functions (positive and
negative arguments for the latter) are generated by our Generalized
Exponential Function (GEF for short) of this Section, so that in a
sense GEF is simpler and more fundamental than boundary and initial/
terminal conditions.

GEF also emphasizes the phase constants and hence phase changes,
including discontinuous ones. The Schrodinger wave function in this
example emphasizes only continuity of the wave function and its
derivative, which is less general. Remember also the Israeli (Ben-
Gurion U.) paper that I cited in one of my last few postings which
indicates that points can usefully be incorporated into QM.

Osher Doctorow

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