and that
. We want to determine the condition
under which 
:
If the linear combination 
is normalized, we must
have 
.
.
According to the uncertainty principle,
If we plug in the numbers, we get 
. Of course, p = mv so 
. For an electron, this gives 
. Needless to say, this is a very
large
error in the speed. For comparison, a free electron at room
temperature would have a speed of approximately 
so the error imposed by a measurement of
position of the given accuracy is greater than typical
thermal velocities.
Schrödinger's equation is therefore
If the problem is separable, we should be able to write the
wavefunction in the form 
.
Making this substitution in Schrödinger's equation, we get
Divide both sides of the equation by 
and
rearrange:
The first parenthesis involves only terms which depend on x, the second only terms which depend on y. Since they add to a constant (E), it follows that each parenthesis must itself be a constant. We get two separate one-dimensional Schrödinger equations:
with 
.
The difference in energy between two adjacent energy levels is
(using the above
formula with n=1). The next line would have an energy
proportional to 
which is 
bigger than the energy of the first line, so the next
line in the spectrum should be found at 
.
so in our case,
. All operators corresponding
to observable are Hermitian. Applying this property to the
expectation value, we get
The two inner products are the same, so we are saying that the expectation value must be equal to its complex conjugate. The only way that can be true is if the imaginary part is zero, so the expectation value is real.