Time independent Schrdinger equation

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Summary of main points so far..
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A quantum solution to some 19th century
problemsFirst Light as particles..
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Black body Radiation Plancks quantum hypothesis
Planck was able to show that his formula would
result if the thermal radiation could only
interchange energy with the cavity in discrete
packets of magnitude h? called quanta. In this
case the classical equipartition of energy fails
the mean energy of each E.M. mode falls below kBT
when h? gt kBT.
Rayleigh-Jeans law
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The Photoelectric Effect
Millikan (1916) found that Kmax did not
depend on the intensity of the light Kmax was
proportional to the frequency of the light For
a particular metal there is a cut-off frequency,
?0, below which no electrons are emitted.
All of these features may be understood in terms
of the particle nature of light Einsteins
photons
Lithium metal
?0
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Einsteins view of photoelectric effect
One electron is emitted by absorbing one
photon Electrons have to overcome a potential
inside the metal before they can escape. This is
called the work function W The minimum photon
energy that can eject an electron is h?0 W
For ? gt ?0 electrons will have a surplus kinetic
energy, so that
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Compton Effect
In 1923 A.H.Compton studied the scattering of
X-rays from a carbon target. He found that the
wavelength of the X-rays was shifted to larger
values. The wavelength shift depended on the
angle through which the X-rays were scattered.
This is what we would expect if the X-ray photons
are scattered inelastically from the loosely
bound electrons in the carbon. The scattered
photon loses energy and changes momentum.
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Compton effect contd
Conservation of momentum Conservation of total
energy
p
?hk
?
Gives
?hk
?C h/mec is the Compton wavelength.
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Bremsstrahlung Spectra
When high energy electrons are fired at a metal
surface they are rapidly decelerated and
therefore radiate E.M. radiation.
There is a sharp cut-off at short wavelengths,
corresponding to the situation where all the
kinetic energy of the electron is converted into
a photon. The maximum photon energy is
therefore Here V is the accelerating potential of
the electrons.
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Second, particles as waves..
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de Broglies Hypothesis
In 1924 Louis de Broglie proposed that, just as
light possesses both wave-like and particle-like
properties, so all moving particles should
possess wave-like properties. In particular any
particle with a momentum p has an associated
wavelength ?, such that
Confimation of this remarkable hypothesis came
from experiments on the diffraction of electrons
by crystal lattices.
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electron diffraction
Davisson Germer (1928) demonstrated the
diffraction of low energy electrons from the
surface of a single crystal of nickel.

n? Dsin?
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Making a wavepacket
..add a number of waves with a spread in
wavenumbers.
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Relation between ?k and ?x
This distribution of k values leads to this
wavepacket.
The theory of Fourier transforms shows that ?k
and ?x are inversely related
Group velocity of the packet
..just like a particle
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Heisenbergs Uncertainty Relations - 1
From Fourier Transforms
but de Broglies relation
so
therefore
i.e. we cannot reduce the uncertainty in position
without increasing the uncertainty in the
momentum of a particle
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Time domain relation between ?? and ?t
This distribution of ? values leads to this
wavepacket.
The theory of Fourier transforms shows that ?t
and ?w are inversely related
So that )t )T . 1
i.e. we cannot reduce pulse width without
increasing spread of frequencies
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Heisenbergs Uncertainty Relations - 2
But from Plancks relation
therefore
i.e. we cannot reduce the uncertainty in time
without increasing the uncertainty in the energy
of a particle
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Interference with particles
slit widtha slit separationd
Usual double slit arrangement.
Weak source of particles only
one particle in the apparatus at a
time. Individual particles are detected on the
screen, but all the individual events make up a
diffraction pattern.
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Schrödingers wave equation
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Operators and Observables
An operator, , operates on a
wavefunction, , to produce an observable
, as well as returning the wavefunction,
unchanged.
Momentum operator
Total energy operator
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The wave function
Wave properties Particle properties
Momentum p k
Energy E T
We can extract the momentum from the
wavefunction by the operation
Similarly we can extract the energy by
operating with
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Schrödingers wave equation
total energy kinetic energy potential
energy E p2/2m V
In operator form Rearranging we get
Schrödingers equation (one dimensional version)
Note complex equation so needs complex wave
function
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Interpretation of the wavefunction
Probability of finding a particle in range
x to x dx at time t is
P(x,t) dx. ? R (x,t)2 dx Normalise
probabilities
(one particle)
i.e.
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Time independent Schrödinger equation
Separation of variables Insert into
Schrödinger equation
Divide through by
constant, E
LHS RHS Only depends on x Only depends
on t
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T.I.S.E. contd
RHS becomes
Integrating LHS becomes
Time-independent Schrödinger equation
Solution Probability density
is independent of time
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Example of T.I.S.EParticle in a box infinite
potential well
For x ? a
Boundary condition u(x) 0 for x gt a
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Summary of solutions
u(x)Asin(kx)
u(x)Bcos(kx)
Boundary condition u(x) 0 for x gt a So
sin(kx) 0 for x a, i.e. kam?
(m any integer)
or kan? /2
(n2,4,6,8) or cos(kx) 0 for x a.
i.e. kan? /2 ( n 1,3,5,7)



(for all values of n)
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Symmetry and Parity
When the potential has inversion symmetry,
ie V (-x) V (x),
the
eigenfunctions always have a definite parity
If wavefunction has
even parity, i.e. a parity of 1 (true
for n odd) If
wavefunction has odd parity, i.e. a parity
of -1 (n even)