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Lect34_HeisenbergPicture

# Lect34_HeisenbergPicture - Lecture 33 Hilbert Space...

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Lecture 33: Hilbert Space Transformations: The Heisenberg Picture PHY851 Quantum Mechanics I Fall, 2008 M.G. Moore

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Real-Space transformations Transformations in Real space include: Translation Boost Rotation Parity reversal We can treat them as either Active or Passive transformations: Active: actual change of the state of the system Apply unitary transformation operators to state vectors only Passive: the same state viewed from a different frame of reference Apply unitary transformation operators only to observables Equivalence: a passive transformation is equivalent to an active transformation in the opposite direction E.g. moving the coordinate origin to the left is equivalent to shifting the wavefunction to the right Be sure not to transform both the states and the observables at the same time Transformations are either cancelled or doubled
Hilbert Space transformations On the other hand, there may be reasons to transform both the states and the observables at the same time If both are transformed via the same operator U , then no physical prediction will be altered by the transformation: Proof: Let O be any observable Let: Let: We see that: UOU O = ψ ψ U = ψ ψ = O O = ψ U + ( ) UOU + U ψ ( ) ψ ψ U UOU U = ψ ψ O = O = 1 = U U

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Time evolution as a Unitary transformation This approach becomes most powerful if we recognize that the time propagator is a unitary operator We have: Where for time-independent Hamiltonians: For the most part, we will assume this is valid But, more generally , U ( t , t 0 ) is the solution to: Which to be true for any initial state requires: Subject to the initial condition: Note that U ( t , t 0 ) must be unitary, since it preserves the normalization of the state.
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