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CH301 Chapter 12 Notes: Part 2:
Solution to the
Schrödinger Equation: The Particle in a 1Dimensional Box
<
n
(x) = (2/L)
1/2
sin(n
3
x/L)
This equation summarizes a set of n sinusoidal curves:
<
1
± <
2
± <
3
± <
4
± <
5
………
Allowed energies of a particle in a 1D box:
E
n
= n
2
h
2
8mL
2
Each solution has its own specific energy:
E
1
±
E
2
±
E
3
±
E
4
±
E
5
………….
n is known as a QUANTUM NUMBER.
In this case, n is a positive, nonzero integer.
Any other value of n would not work  it would not be a solution to the Schrödinger equation!
What happens to E for any value of n if we let m or L increase?
Figure 12.14 shows the energies, wavefunctions and square of the wavefunctions for the
first three solutions (n=1, n=2, n=3).
In class we'll discuss the third one in detail.
Interpreting the solution for a particle in a 1D box (Figure 12.14):
Each wavefunction is equal to
zero at x = 0 and x = L (as required by the boundary conditions)
Note the energy level ‘ladder’
 QUANTIZED energies.
Sketch the plots of
to
<
3
and
<
²
2
here and add the comments/notes:
Remember
<
2
gives the idea of WHERE the particle is likely to be!
How does this compare to what you would expect for a macroscopic object?
What do I need to know?
Be able to sketch and interpret any of the first 8 or so solutions: e.g.,
<
²
±
or
<
³
2
….
Where are the NODES. What is a Node?
Where will you be most/least likely to find the particle? What does
<
2
mean?
Energy needed to jump from one level to another: (Figure 12.14)
'
E = (n
2
2
n
1
2
)
h
2
8mL
2
We can calculate the energy to go from
<
n
1
to level
<
n
2
:
where n
1
and n
2
are two different values of n. We see that the gaps
themselves,
'
E,
are
also
inversely proportional to L (actually L
2
) and m.
Example: A ball mass 0.5g in a 1D box 2m long: What energy is needed to go from n=4 to n=7 ?
Zero Point Energy: Quantum Weirdness
What about n=0?
<
0
does NOT exist! The smallest allowed value of n is 1. The energy of a
particle in a 1D box can
never
be zero! To calculate
ZPE
use
n=1
and your values of m and
L.
Example: Assume we confine all of these in 1D boxes
100cm long:
 an electron
 a marble
 a bowling ball
Which do you think has the smallest ZERO POINT ENERGY?
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View Full Document What do I need to know?
Be able to sketch the energy level diagram and interpret it: (what happens as we change n,L)
Be able to do calculations of values of E for a particular wavefunction given n,m,L
Be able to calculate
'
E for a particular wavefunction given two values of n, and m,L
Be able to calculate the ZPE if given m,L (obviously you know what n is!!! )
Also the sort of qualitative examples we’ve done.
Always remember that you should get
very
tiny numbers if your object is macroscopic. Why?
The Schrodinger Equation for the H atom
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This note was uploaded on 11/02/2010 for the course CH 50985 taught by Professor Sarasutcliffe during the Fall '10 term at University of Texas at Austin.
 Fall '10
 SaraSutcliffe

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