Quantum mechanics review

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Quantum mechanics review
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• Reading for week of 1/28-2/1
• Chapters 1, 2, and 3.1,3.2
• Reading for week of 2/4-2/8
• Chapter 4

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Schrodinger Equation (Time-independent)
where
The solutions incorporate boundary conditions and
are a family of eigenvalues with increasing
energy and corresponding eigenvectors with an
increasing number of nodes. The solutions are
orthonormal.
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Physical properties Expectation values
Dirac notation or bra-ket notation
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Physical properties Hermitian Operators
Real Physical Properties are Associated with
Hermitian Operators Hermitian operators obey the
following
The value ltAgtmn is also known as a matrix
element, associated with solving the problem of
the expectation value for A as the eigenvalues of
a matrix indexed by m and n
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Zero order models Particle-in-a-box atoms,
bonds, conjugated alkenes, nano-particles Harmoni
c oscillator vibrations of atoms Rigid-Rotor
molecular rotation internal rotation of methyl
groups, motion within van der waals
molecules Hydrogen atom electronic structure
Hydrogenic Radial Wavefunctions
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Particle-in-a-3d-Box
V(x) 0 0ltxlta V(x) 8 xgta x lt0 b? y c ? z
V(x)
x
a
nx,y,z 1,2,3, ...
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Particle-in-a-3d-Box
V(x) 0 0ltxlta V(x) 8 xgta x lt0 b? y c ? z
V(x)
x
a
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Zero point energy/Uncertainty Principle
In this case since V0 inside the box E
K.E. If E 0 the p 0 , which would violate
the uncertainty principle.
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Zero point energy/Uncertainty Principle
More generally Variance or rms
If the system is an eigenfunction of then
is precisely determined and there is no
variance.
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Zero point energy/Uncertainty Principle
If the commutator is non-zero then the two
properties cannot be precisely defined
simultaneously. If it is zero they can be.
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Harmonic Oscillator 1-d F-k(x-x0)
Internal coordinates Set x00
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Harmonic Oscillator Wavefunctions
Hermite polynomials
V quantum number 0,1,2,3
Hv Hermite polynomials Nv Normalization
Constant
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http//hyperphysics.phy-astr.gsu.edu/hbase/quantum
/hosc5.htmlc1
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Raising and lowering operators Recursion
relations used to define new members in a family
of solutions to D.E.
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Rotation Rigid Rotor
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Rotation Rigid Rotor
Wavefunctions are the spherical harmonics
Operators L2 ansd Lz
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Degeneracy
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Angular Momemtum operators the spherical harmonics
Operators L2 ansd Lz
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Rotation Rigid Rotor
Eigenvalues are thus
l 0,1,2,3,
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Lots of quantum mechanical and spectroscopic
problems have solutions that can be usefully
expressed as sums of spherical harmonics. e.g.
coupling of two or more angular momentum plane
waves reciprocal distance between two points in
space Also many operators can be expressed as
spherical harmonics
The properties of the matrix element above are
well known and are zero unless -mMm
0 lLl is even
Can define raising and lowering operators for
these wavefunctions too.
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The hydrogen atom
Set up problem in spherical polar coordinates.
Hamiltonian is separable into radial and angular
components
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n the principal quantum number, determines
energy l the orbital angular momentum
quantum number l n-1, n-2, ,0 m the
magnetic quantum number -l, -l1, , l
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