The Heisenberg picture

The Heisenberg picture (HP) is a formal trick, shifting the dynamical time- dependence of the Schrodinger-Dirac equation (24.1) from the state vectors onto the operators. The following analogy may help explain what is going on. Consider yourself standing on the ground in a fairground observing a moving merry-go-round carrying a friend. In this analogy, your friend is the analogue of the Schrodinger wavefunction Фг: they are moving whilst you, the observer, is at rest in the laboratory. This scenario is analogous to the SP.

Suppose now you jumped onto the moving merry-go-round and sat next to your friend. From this new perspective the merry-go-round and your friend would appear to be at rest relative to you, whilst the rest of the universe would appear to be moving in a direction opposite to the one your friend appeared to have before you joined them. This scenario is the analogue of the HP. The point is, both pictures describe the same reality.

To see how the HP is obtained from the SP, define the HP state = Фго. Then

the SP state Фг is related to Фн by

where the subscript H refers to the Heisenberg picture.

To understand the effect of the transition from the SP to the HP on the observables of the theory, that is, the operators we use to squeeze out physical predictions from the states, consider some observable Ot in the SP, an observable that may have some explicit time dependence. The standard rule in QM is that the expectation value (Ot), or average of many separate, ensemble measurements of the observable О relative to the state Фг at time t, is given by[1]

Using (24.6), we readily find it to be given by

where OHt is the HP operator defined by OH,t = U+^OtUm, where U+ is the adjoint operator of Um (adjoint operators are discussed in the Appendix).

The fact that we can use the HP should give anyone who believes in the reality or objectivity of wavefunctions serious food for thought. Clearly, in the HP, all the dynamical evolutionary processes are now associated with the measuring apparatus and quantum states are frozen in time. Therefore, time is not an intrinsic attribute of quantum states: it depends how you choose to look at it.

  • [1] This assumes Ф, is not the zero vector in the Hilbert space. The denominator is there to‘normalize’ the probabilities.
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