The Schrodinger and Heisenberg representations

Consider again the time evolution equations (2.1)—(2.3). If A is an operator representing a physical observable, the expectation value of this observable at time t is A t = ( h G) 1 1 1 (t))- We can express this same quantity differently. Define 

Obviously, is simply 4 (Z = 0) and is by definition time independent. Equation (2.62) is a unitary transformation on the wavefunctions and the operators at time t. The original representation in which the wavefunctions are time dependent while the operators are not, is transformed to another representation in which the operators depend on time while the wavefunctions do not. The original fonnulation is referred to as the Schrodinger representation, while the one obtained using (2.62) is called the Heisenberg representation. We sometimes use the subscript S to emphasize the Schrodinger representation nature of a wavefunction or an operator, that is. 

Either representation can be used to describe the time evolution of any observable quantity. Indeed 

Note that the invariance of quantum observables under unitary transformations has enabled us to represent quantum time evolutions either as an evolution of the wavefunction with the operator fixed, or as an evolution of the operator with constant wavefunctions. Equation (2.1) describes the time evolution of wavefunctions in the Schrodinger picture. In the Heisenberg picture the wavefunctions do not evolve in time. Instead we have a time evolution equation for the Heisenberg operators  

Equation (2.66) is referred to as the Heisenberg equation of motion. Note that it should be solved as an initial value problem, given that H(f = 0) = A. In fact, Eq. (2.62b) can be regarded as the formal solution of the Heisenberg equation (2.66) in the same way that the expression (Z) = = 0) is a formal solution 

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