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The Mathematics of Quantum
Mechanics
2. Unitary and Hermitian Operators
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Measurement in Quantum Mechanics
Measuring is equivalent to decomposing down
the system state to its basis states. What are
the basis states for a specific measurement?
What values are obtained in the
measurement?
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The Rotation Group
Rotation Operator
Group Properties
The mathematical
generator of the group
If the change in the function is known as a result of an infinitesimal
rotation, the integrated result of applying any rotation operator can
be calculated by this infinitesmal change
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The Angular Momentum Operator
The generator of the rotation group is used to construct a
physically meaningful operator
The result of operating an angular momentum operator on a
function is equivalent to a derivation by 
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The Eigenfunctions and Eigenvalues
The eigenvalue equation
The eigenfunctions and eigenvalues are obtained as a result of
solving the differential equation
 The eigenfunctions of the angular momentum operator are the
basis states of the measuring of the angular momentum
 The eigenvalues of the angular momentum operator are the
results obtained by measurement
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The Link Between the Group Operators and
Generator Operators
1. The eigenfunctions are identical
2. The link between the eigenvalues
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The rotation operator represents a geometrical action that
preserves the normalization, and therefore 1 = || 2 (Unitary
Operator)
The angular momentum operator represents a measurable
physical quantity, and therefore all the eigenvalues are real
(Hermitian Operator)
The propagation in Time
The Schrödinger equation describes how the system state
changes in time:
The second postulate of a free particle
The second postulate of a two-dimensional rigid rotor
Which operator generates the change in time?
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The Evolution Group
The Evolution Operator
The Group Generator:
The Group properties
The Schrödinger time-dependent equation:
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The Hamiltonian - the Energy Operator
1. The dispersion ratio in the second postulate determines the
eigenvalue of the evolution operator:
2. An evolution eigenfunction is also a Hamiltonian eigenfunction
(and vice versa). Also, the stationary Shrödinger equation should
be fulfilled (according to the link between the group operators
and the generator):
The generator of the evolution operator represents the measurement of
energy !
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An Example: a Free Particle on a Ring
Inserting the appropriate Hamiltonian:
From classic mechanics E=Lz2/2I
Equation of the eigenvalue:
Eigenfunctions:
Obviously
But also:
Eigenvalues:
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is fulfilled
The Significance of the Hamiltonian in Chemistry
1. Finding the basis state - the eigenfunction with the
lowest eigenvalue is the most energetically stable
state of a chemical system
2. Sperctroscopy - measuring of energy states. The basis
states of the measurment are the eigenfunctions of the
Hamiltonian, and the measured values are the
appropriate eigenvalues.
3. Dynamical calculations - the eigenfunctions of the
Hamiltonian are also eigenfunctions of the evolution
operator
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