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Review Three 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 exponential’s 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 Heisenberg Operator: Equation of motion for Heisenberg Operators At t=0: Complicated when since
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Interaction Picture Let’s divide the Hamiltonian into two parts: Usually H 0 is a soluble problem. What are the effects of H 1 ? 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 normalizableHermite series must terminate
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Matrix Notation Let: Define inner product: Orthonormality of wave functions gives: Complete solution:where Since Schrödinger equation is linear and the set of eigenfunctions is complete:
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Energy Quantization from Commutation Relationships Either or
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Raising and Lowering Operators Combining red equations in another way: Thus, eitheror Thus the operator: lowers the state by one Likewise the operator: raises the state by one Define dimensionless lowering 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.
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