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A very elementary approach to Quantum mechanics

There was a time when newspapers said that only
twelve men understood the theory of
relativity. I do not believe that there ever
was such a time... On the other hand,
I think it is safe to say that
no one understands quantum mechanics
R.P. Feynman
The Character of

Physical Law (1967)
? lets approach some aspects of qm anyway
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Experimental facts
Light has wave (interference) and particle
properties
Plot from
Existence of photons
Radiation modes in a hot cavity provide a test
of quantum theory
Frequency
Plancks const.
Energy of the quantum
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Energy of a free particle
where
Consider photons
and
with
or
Dispersion relation for light
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Electrons (particles) have wave properties
Figures from
de Broglie
applicable for particles
Today LowEenergyElectronDiffraction standard
method in surface science
LEED Fe0.5Zn0.5F2(110) 232 eV
top view (110)-surface
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Implications of the experimental facts
Electrons described by waves
Wave function (complex for charged particles
like electrons)
Probability to find electron at (x,t)
Which equation describes the temporal evolution
of
Schroedinger equation
Cant be derived, but can be made plausible
Lets start from the wave nature of, e.g., an
electron
Erwin Schroedinger
and take advantage of
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In complete analogy we find the representation of
E
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Schroedinger equation for 1 free particle
Hamilton function of classical mechanics
HE total energy of the particle
1-dimensional
In 3 dimensions
where
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Schroedinger equation for a particle in a
potential
Classical Hamilton function
Time dependent Schroedinger equation
Hamilton operator
If
independent of time like
only stationary Schroedinger equation has to be
solved
Proof
Ansatz
(Trial function)
Stationary Schroedinger equation
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Solving the Schroedinger equation (Eigenvalue
problem)
Solution requires
-Normalization of the wave function according
Physical meaning probability to find the
particle somewhere
in the universe is 1
-Boundary conditions of the solution
have to be continuous when merging piecewise
solutions
Note boundary conditions give rise to the
quantization
Particle in a box
Eigenfunctions
Eigenenergies
Quantum number
x
Details see homework
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Heisenbergs uncertainty principle
It all comes down to the wave nature of particles
Wave function given by a single wavelength
Momentum p precisely known, but where is the
particle position
-P precisely given
-x completely unknown
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Wave package
Fourier-analysis
Fourier-theorem
In analogy