lesson 5 particle in a box - stanford university · pdf fileso, if is a solution, ... to...
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2.3 Particle in a box
Slides: Video 2.3.1 Introduction to the particle in a box
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The particle in a box
David MillerQuantum mechanics for scientists and engineers
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2.3 Particle in a box
Slides: Video 2.3.2 Linearity and normalization
Text reference: Quantum Mechanics for Scientists and Engineers
Section 2.4 – 2.5
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The particle in a box
Linearity and normalization
Quantum mechanics for scientists and engineers David Miller
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Linearity and Schrödinger’s equation
We see that Schrödinger’s equation is linear
The wavefunction appears only in first orderthere are no second or higher order terms
such as 2 or 3
So, if is a solution, so also is athis just corresponds to multiplying both sides by the constant a
2
2
2V E
m
r
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Normalization of the wavefunction
Born postulatedthe probability of finding a particle near a point r is
Specifically let us define as a“probability density”
For some very small (infinitesimal) volume d 3r around r
the probability of finding the particle in that volume is
P r 2
r
P r
3P dr r
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Normalization of the wavefunction
The sum of all such probabilities should be 1So
Can we choose so that we can use as the probability density
not just proportional to probability density?Unless we have been lucky
our solution to Schrödinger’s equation did not give a so that
3 1P d r r
r 2 r
r 2 3 1d r r
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Normalization of the wavefunction
Generally, this integral would give some other real positive number
which we could write aswhere a is some (possibly complex)
numberThat is,
But we know that if is a solution of Schrödinger’s equation
so also is
21/ a
2 32
1da
r r
r
a r
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Normalization of the wavefunction
Soif we use the solution instead of
then
and we can use as the probability density, i.e.,
would then be called a“normalized wavefunction”
N a
2 3 1N d r r
2NP r r
2N r
N r
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Normalization of the wavefunction
So, to summarize normalizationwe take the solution we have obtained from Schrödinger’s wave equation
we integrate to get a number we call
then we obtain the normalized wavefunction for which
and we can use as the probability density
2 r
21/ a
N a
2 3 1N d r r
2N r
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Technical notes on normalization
Note that normalization only sets the magnitude of a
not the phasewe are free to choose any phase for a
or indeed for the original solution a phase factor is just another number by which we can multiply the solution
and still have a solution
exp i
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Technical notes on normalization
If we think of space as infinitefunctions like , , and
cannot be normalized in this wayTechnically, their squared modulus is
not “Lebesgue integrable”They are not “L2” functions
This difficulty is mathematical, not physical It is caused by over-idealizing the mathematics to get functions that are simple to use
sin kx cos kz exp i k r
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Technical notes on normalization
There are “work-arounds” for this difficulty1 - only work with finite volumes in actual problems
this is the most common solution2 - use “normalization to a delta function”
introduces another infinity to compensate for the first one
This can be donebut we will try to avoid it
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2.3 Particle in a box
Slides: Video 2.3.4 Solving for the particle in a box
Text reference: Quantum Mechanics for Scientists and Engineers
Section 2.6 (first part)
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The particle in a box
Solving for the particle in a box
Quantum mechanics for scientists and engineers David Miller
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Particle in a box
We consider a particle of mass mwith a spatially-varying potential V(z) in the z direction
so we have a Schrödinger equation
where E is the energy of the particleand (z) is the wavefunction
22
22d z
V z z E zm dz
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Particle in a box
Suppose the potential energy is a simple “rectangular” potential well
thickness LzPotential energy is constant inside
we choose there rising to infinity at the walls
i.e., at and We will sometimes call this
an infinite or infinitely deep (potential) well
Ener
gy0V
0z zz L
0z zz LzL0V
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Particle in a box
Because these potentials at and at are infinitely highbut the particle’s energy E is
presumably finitewe presume there is no possibility of finding the particle outside
i.e., for orso the wavefunction is 0 there
so should be 0 at the walls
Ener
gy
zL
0z zz L
0z zz L
0z zz L
0V
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Particle in a box
With these choicesinside the well
the Schrödinger equation
becomes
with the boundary conditions and
Ener
gy
zL0z zz L
0V
22
22d z
V z z E zm dz
22
22d z
E zm dz
0 0 0zL
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Particle in a box
The general solution to the equation
is of the form
where A and B are constants
andThe boundary condition
means because
Ener
gy
zL0z zz L
0V
22
22d z
E zm dz
sin cosz A kz B kz
22 /k mE 0 0
0B cos 0 1
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Particle in a box
With now and the condition
kz must be a multiple of , i.e.,
where n is an integer
Since, therefore,
the solutions are
with
Ener
gy
sinz A kz 0zL
22 / / zk mE n L
2 2
2kEm
sinn nz
n zz AL
22
2nz
nEm L
1E1n
2E2n
3E3n
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Particle in a box
We restrict n to positive integers for the following reasonsSince for any real
number athe wavefunctions with negative nare the same as those with positive n within an arbitrary factor, here -1
the wavefunction for is trivialthe wavefunction is 0 everywhere
Ener
gy
1E
2E
3E
1n
2n
3n
1, 2,n
sin sina a
0n sinn n
z
n zz AL
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Particle in a box
We can normalize the wavefunctions
To have this integral equal 1choose
Note An can be complexAll such solutions are arbitrary
within a unit complex factorConventionally, we choose An
real for simplicity in writing
Ener
gy
1E
2E
3E
1n
2n
3n 2 22
0
sin2
zLz
n nz
Ln zA dz AL
2 /n zA L
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Particle in a box
Ener
gy
1E
2E
3E
1n
2n
3n
2 sinnz z
n zzL L
22
2nz
nEm L
1, 2,n
zL0
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2.3 Particle in a box
Slides: Video 2.3.6 Nature of the particle-in-a-box solutions
Text reference: Quantum Mechanics for Scientists and Engineers
Section 2.6 (second part)
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The particle in a box
Nature of particle-in-a-box solutions
Quantum mechanics for scientists and engineers David Miller
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Eigenvalues and eigenfunctions
Solutionswith a specific set of allowed values
of a parameter (here energy)eigenvalues
and with a particular function associated with each such valueeigenfunctions
can be called eigensolutions
2 sinnz z
n zzL L
22
2nz
nEm L
1, 2,n
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Eigenvalues and eigenfunctions
Heresince the parameter is an energy
we can call the eigenvalueseigenenergies
and we can refer to the eigenfunctions as theenergy eigenfunctions
2 sinnz z
n zzL L
22
2nz
nEm L
1, 2,n
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Degeneracy
Note in some problems it can be possible to have more than one eigenfunction with a given eigenvalue
a phenomenon known as “degeneracy”
The number of such states with the same eigenvalue is called
“the degeneracy”of that state
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Parity of wavefunctions
Note these eigenfunctions have definite symmetrythe function is the mirror
image on the left of what it is on the rightsuch a function has “even parity”or is said to be an “even function”
The eigenfunction is also even 1n
2n
3n 1n
3n
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Parity of wavefunctions
The eigenfunction is an inverted image the value at any point on the right
of the centeris exactly minus the value at the “mirror image” point on the left of the center
Such a function has “odd parity”or is said to be an “odd function”
1n
2n
3n 2n
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Parity of wavefunctions
For this symmetric well problemthe functions alternate between
being even and oddand all the solutions are either even or oddi.e., all the solutions have a
“definite parity”Such definite parity is common in
symmetric problemsit is mathematically very helpful
1n
2n
3n
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Quantum confinement
This particle-in-a-box behavior is very different from the classical case1 – there is only a discrete set of
possible values for the energy2 – there is a minimum possible
energy for the particlecorresponding to
heresometimes called a
“zero-point energy”
1n
2n
3n
Ener
gy
1E
2E
3E
zL0
1n 22
1 / 2 / zE m L
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Quantum confinement
3 - the particle is not uniformly distributed over the box, and its distribution is different for different energiesIt is almost never found very
near to the walls of the boxthe probability obeys a
standing wave pattern 1n
2n
3n
Ener
gy
1E
2E
3E
zL0
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Quantum confinement
In the lowest state ( ), it is most likely to be found
near the center of the boxIn higher states,
there are points inside the box where the particle will never be found
1n
2n
3n
Ener
gy
1E
2E
3E
zL0
1n
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Quantum confinement
Note that each successively higher energy
state has one more “zero” in the eigenfunction
This is very common behavior in quantum mechanics
1n
2n
3n
Ener
gy
1E
2E
3E
zL0
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Energies in quantum mechanics
In quantum mechanical calculationswe can always use Joules as units of energy
but these are rather largeA very convenient energy unit
which also has a simple physical significanceis the electron-volt (eV)
the energy change of an electron in moving through an electrostatic potential change of 1V
Energy in eV = energy in Joules/ee – electronic charge (Coulombs)
191.602 10 J
191.6021 76 565 10 C
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Orders of magnitude
E.g., confine an electron in a 5 Å (0.5 nm) thick box
The first allowed level for the electron is
The separation between the first and second allowed energies ( )
is which is a characteristic size of major
energy separations between levels in an atom
22 10 191 / 2 / 5 10 2.4 10 1.5J eVoE m
2 1 13E E E 4.5eV
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