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Lecture 10 - Finite Square Well.pdf
Lecture_10_-_Finite_Square_Well.pdf
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Lecture 10 - Finite Square Well.pdf-Today (1) Revi...
Lecture_10_-_Finite_Square_Well.pdf-Today (1) Review of time dependent
Lecture 10 - Finite Square Well.pdf...
Lecture_10_-_Finite_Square_Well.pdf-Today (1) Review of time dependent
Page 1
1
Today
(1) Review of time dependent Vs time independent
Schrodinger eqtn
(2) When is it appropriate to use the infinite square well
(3) The finite square well and its solution
In this course, the reason why we always solve the
simpler
time independent
(TISE) Schrodinger eqtn is because
V(x,t) is ALWAYS time independent. In a more advance course,
you learn to deal with time dependence in the potential.
The infinite square well is the SIMPLEST physics system…
Simpler than the finite square well for instance
Page 2
O.k., how do we go from the time dependent Schrodinger eqtn to the time
independent one (TISE)?
We need to use separation of variables!
We first assume a separable soln form :
Plug it in and we get:
We divide both sides by
and get 2 eqtns:
Only depends on x
Only depends on t
How can something depending only on
x
and something depending only on
t
be equal to one another? Only if they are both equal to a constant…lets
call that constant
E.
Of course, it’ll become obvious quickly that the separation
constant actually IS the total energy.
Separation constant
Page 3
‐
Note that the beautiful separation only happened because the potential
was
V(x)
and NOT
V(x,t).
VERY IMPORTANT!!
‐
Now we have the time independent Schrodinger equation:
And it looks like the time dependence is really easy to solve:
The solution is:
‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
O.k. so total solution is always
K.E. + P.E. = Total Energy
and we read that:
Hummm, more generally the soln is actually :
WAIT !!!!
:
What just happened here…why a sum? What are these c’s in front of Psi?
Lets do a
CLICKER QUESTION
Total energy
????
Page 4
4
Warm-up question:
In the classically forbidden region for a finite potential well,
the physically meaningful solution to the Schrodinger
equation is
(
is always a real positive constant)
a. a cosine function (cos(x/
) )
b. the sum of sine and cosine functions
c. an increasing exponential
function( e
+x/
)
d. a decaying exponential
function( e
-x/
)
e. this question makes no sense
ans. d.
classically forbidden region,
E total less than potential energy V(x)
x
(x)
Page 5
5
one place where this model is “reasonably” accurate
Nanotechnology: how small does an object have to be
before movement of electrons starts to depend on size
and shape due to quantum effects?
Solving the Schrodinger equation for simple model
system (square well).
Easiest case of electron
confined to solve diff. equation.
Form of solutions
simpler, easier to understand basic ideas of QM.
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