Deriving Heisenberg's Equation
The expectation value of an observable A, which is a Hermitian linear operator, for a given state is given by:
In general where is the time evolution operator. For an elementary derivation, we will take Hamiltonian to commute with itself at different times, and further, be independent of time, in which case it simplifies to:
Where H is the Hamiltonian and ħ is Planck's constant divided by . It follows that:
We define:
It follows:
We differentiated according to the product rule and noted that is the time derivative of the operator we started with. The last passage is valid since commutes with H.
From this results the Heisenberg equation of motion:
is the commutator of two operators and defined as := XY − YX.
Now, using the operator identity:
One obtains for an observable A:
This relation also holds for classical mechanics, due to the relationship between Poisson bracket and commutators, which is:
Hence, in classical mechanics:
The expression of A(t) is the Taylor expansion on t = 0.
Read more about this topic: Heisenberg Picture
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