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MODULE 1
In classical mechanics we define a STATE as
“The specification of the position and velocity of all the particles present, at
some time, and the specification of all the forces acting on the particles.”
Then Newton’s (or any other) classical equations of motion allow us to
determine the state of the system at any future time.
In quantum theory the specification of the state of a system at a given time is
provided by its wavefunction (Postulate 1).
In order to predict the state of a sub-microscopic system at another time we
need a quantum equivalent of Newton’s law that tells us how the
wavefunction changes with time.
This is the Schrödinger equation, which we present as a Postulate.
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POSTULATE 3
“The wavefunction Y(r1, r2, …, t) changes in time according to
Y
ˆ
HY  i .
t
where Ĥ is the hamiltonian (total energy ) operator".
This is the time-dependent equation that Schrödinger introduced
in 1926.
Here we are postulating it.
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In the case where we have a single particle and the sole spatial
variable is the dimension x, we can substitute for the
hamiltonian by the expression found earlier, thus

Y  2  2
i
 
 V ( x)  Y
2
 t  2m  x

This time-dependent Schrödinger equation can be separated into
a pair of equations, one spatial and one temporal.
(See Math Module A and Barrante ch 6)
d 2

 V ( x)  E
2
2m dx
d
i
 E
dt
time independent
time dependent
2
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Integrating the temporal equation gives
   0 exp(
iEt
)
and so the complete wavefunction can be written as
 iEt 
Y ( x, t )   ( x) exp  



where the constant of integration has been incorporated into the
normalization constant for .
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The time-independent Schrödinger equation has the form of a
standing wave equation and if all we are interested in are
spatial dependences, then this is the one for us.
If the potential energy is independent of time, and the system has
energy, E, we can construct the time-dependent wavefunction
from the time-independent one simply by multiplication by
e
 iEt /
The Euler relationship allows us to write
eiEt /  cos( Et / )  i sin( Et / )
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Thus the time dependence of Y is actually a phase modulation
The wave function rotates in the complex plane from real to imaginary and back
to real with a frequency equal to
E/
As time proceeds, the cosine will oscillate between 0 and 1, and the sine from 1
to 0.
eiEt /  cos( Et / )  i sin( Et / )
When the cosine is 0 the sine will be 1 and the second term overall will equal –i,
and the total wave function will be imaginary.
When the cosine is 1 the imaginary term vanishes and the total wavefunction
becomes real.
Thus as time goes by the total wave function will alternately change from real to
imaginary, and so on.
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Even though the amplitude of Y oscillates between real and
imaginary values, the product Y*Y is real and remains
constant, since
Y * Y   * e
iEt /
 e
 iEt /
   *
These systems are stationary states.
They have a specific, precise energy (E) and the potential energy
is not time dependent.
Their wavefunctions flicker between the real and the imaginary,
but the probability density is constant with time.
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We have just used our recipe for constructing the hamiltonian (total energy)
operator.
Now we return to the spatial Schrödinger equation and put it into the form of an
operator equation
2


d2


V
(
x
)

  E
2
 2m dx

the quantity in the brackets is recognized as the hamiltonian operator Ĥ
The spatial Schrödinger equation above is a special equation. It states that the
result of carrying out the (hamiltonian) operation on the wavefunction is the
function itself, multiplied by a constant, in this case the energy E.
When the result of an operation yields a function that is linearly proportional to
the function prior to the operation, then the function is termed an
EIGENFUNCTION of the operator, and the proportionality constant is called
its EIGENVALUE.
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The generic equation
Âf  af
(f is some function and  is an operator)
is an eigenvalue equation with f an eigenfunction and a (a
constant) is the eigenvalue.
Thus the hamiltonian operator operates on the spatial
wavefunction  in an eigenvalue equation called the (timeindependent) Schrödinger equation
Ĥ  E
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The quantity E (the eigenvalue) is the energy observable that
corresponds to the total energy operator Ĥ
This example has demonstrated the method for calculating any
desired observable, i.e., operate on the wave function with the
appropriate operator, then the eigenvalue will represent the
desired quantity.
Eigenvalue equations play a major role in Quantum Chemistry,
but Schrödinger did not invent them; they had been around in
mathematics for a long time.
He adopted the technique when he developed his wave
mechanics.
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Planck hypothesized that an atomic oscillator could exist only in
discrete energy states and as it changed from one state to
another it either emitted or absorbed a quantum of energy equal
to the difference between the two states.
Einstein postulated that light was a quantized form of radiant
energy and Bohr used the quantum concept to explain stationary
angular momentum states in his theory of the H atom.
However the restriction of a physical system to discrete solutions
is not a property of the size of particles, per se.
Submicroscopic particles can undergo motion that is not subject to
quantum restrictions (more on this later).
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Moreover, some macroscopic systems under special conditions
show discrete states of motion that resemble quantization.
Examples are a violin/guitar/etc string, or a stretched drum skin.
These are clamped at certain points where their motion away from
the rest position always necessarily has zero amplitude.
When the string or skin (a 2-dim string) is set into vibrational
motion the equation of motion (from Newton’s second law) is a
second order differential equation that can be separated into
spatial and temporal eigenvalue equations.
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The solutions to the spatial eigenvalue equation form a set of
discrete vibrational frequencies (the fundamental and its
overtones).
This is an example of quantization on a macro-scale that arises
because the zero amplitude fixed points force boundary conditions
on the spatial equation.
This restricts the allowed solutions to integral values of the
fundamental frequency.
It is the application of particular BOUNDARY CONDITIONS that
generates quantization, not the size of the entity.