Review Three Pictures of Quantum Mechanics

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ReviewThree Pictures of Quantum Mechanics
Simple Case Hamiltonian is independent of time

1. Schrödinger Picture Operators are independent
   of time state vectors depend on time.
2. Heisenberg Picture Operators depend on time
   state vectors are independent of time.
3. Interaction pictures Intermediate view both
   state vectors and operators depend on time.

All pictures are equivalent. We will use each at
different times
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Schrödinger Picture
Operators are independent of time. Time
dependence is in wave fucntion
Integrate with respect to time
Re-iterate
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Schrödinger Picture
Continue to re-iterate. If series converges,
then
Note Definition of the exponential of an
operator is the exponentials series expansion of
that operator.
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Heisenberg Picture
States are independent of time. Operators carry
time dependence.
Showing time independence of this definition
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Heisenberg and Schrödinger Operators
Complicated when
Heisenberg Operator
Equation of motion for Heisenberg Operators
At t0
since
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Interaction Picture
Lets divide the Hamiltonian into two parts
Usually H0 is a soluble problem. What are the
effects of H1?
Define
Generally
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What Is Second Quantization?Review of Simple
Harmonic Oscillator
Schrödinger Equation
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Solution to Simple Harmonic Oscillator
Assume
Then
and
where
Wave functions are normalizable
Hermite series must terminate
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Matrix Notation
Let
Define inner product
Orthonormality of wave functions gives
Since Schrödinger equation is linear and the set
of eigenfunctions is complete
Complete solution
where
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Energy Quantization from Commutation Relationships
or
Either
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Raising and Lowering Operators
Combining red equations in another way
Thus, either
or
lowers the state by one
Thus the operator
Define dimensionless lowering operator
raises the state by one
Likewise the operator
Define dimensionless raising operator
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Number Operator
With these definitions
Number operator
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Ground State
Assume that there is a lowest state such that
All other states can be built from the ground
state by repeated applications of the raising
operator
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Heisenberg States
Are stationary in time.
Time development is in the operators
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Step to Second Quantization
Consider the complete set of time independent SHO
Heisenberg states
The relationship between one state and another is
the addition or subtraction of an elemental
excitation (exciton) represented by the creation
operator (raising operator) a and the
destruction operator (lowering operator) a
respectively. Each exciton is represented by an
the operator a and has its own equation of
motion given by
Second quantization is the process of considering
excitations of a system as individual particles
with their own equations of motion.