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NotesSet3(1)

# NotesSet3(1) - PHY4221 Quantum Mechanics I Fall 2004 1.3.4...

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PHY4221 Quantum Mechanics I Fall 2004 1.3.4 Time-dependent Schr¨ odinger Equation Within the Dirac’s formalism of quantum mechanics, the time evolution of an operator ˆ O associated with a dynamical system with Hamiltonian ˆ H is described by the equation d ˆ O dt = ˆ O ∂t + 1 i ¯ h h ˆ O, ˆ H i . (1) Provided that ˆ O does not explicitly depend upon t , i.e. ˆ O ∂t = 0 , (2) we have h ˆ O, ˆ H i = i ¯ h d ˆ O dt ˆ O ( t ) = exp ˆ i ˆ Ht ¯ h ! ˆ O (0)exp ˆ - i ˆ Ht ¯ h ! , (3) where the time t = 0 is of course arbitrary. This is the approach used by Heisenberg to describe the time evolution of a system. Using a constant set of basis vectors , the time evolution of the expectation value of any dynamcial variable of the system in any state | ψ i can be obtained by taking the expectation value with respect to | ψ i of the time-dependent operator ˆ O ( t ): h ψ | ˆ O ( t ) | ψ i = h ψ | exp ˆ i ˆ Ht ¯ h ! ˆ O (0)exp ˆ - i ˆ Ht ¯ h ! | ψ i . (4) This approach of describing time variation is known as the Heisenberg representation . However, Eq.(4) can be interpreted in another way: if we deﬁne the time-dependent state vector as | ψ ( t ) i = exp ˆ - i ˆ Ht ¯ h ! | ψ

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NotesSet3(1) - PHY4221 Quantum Mechanics I Fall 2004 1.3.4...

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