Hamilton Equation and Heisenberg Equation of Motion

[Ang Han Wei]

I was given the (general) following form for the Hamilton and Heisenberg Equations of motion

$\dot{A}$ = $(A, H)_{}$, which can represent the Poisson bracket (classical version)

or

$\dot{A}$ = -i/h[A,H] (Quantum Mechanical commutator).

I was given the general solutin for A(t) = $e^{tL}$A. In addition, L is an operator acting on functions of initial conditions (q,p) in the classical case or quantum mechanical operator in the Hilbert Space such that Lf = $(f, H)_{}$

I am asked to show that A(t) = $e^{tL}$A is the formal solution to $\dot{A}$ = $(A, H)_{}$

I started with the Heisenberg picture in which the obervables are time dependent and differentiated A(t) wrt time t. I got the following:

$\frac{dA(t)}{dt}$ = L$e^{tL}$$\frac{∂L}{∂T}$A + $e^{tL}$$\frac{∂A}{∂t}$

This is where I am stuck as I am unable to find ways to introduce the Hamiltonian H into the equation and show that this is a formal solution to $\dot{A}$ = $(A, H)_{}$

Any help on how i should continue from here will be appreciated!

 

[eljose79]

it is important to carefully analyze and understand the given information before attempting to solve the problem. In this case, it is clear that we are dealing with the Hamilton and Heisenberg equations of motion, which are fundamental equations in classical and quantum mechanics, respectively.

Let's start by breaking down the given equation \dot{A} = (A, H)_{}. This equation represents the Poisson bracket in classical mechanics, where A and H are observables and (A, H)_{}, also known as the Poisson bracket, is a mathematical operation that gives the rate of change of the observable A with respect to time. In quantum mechanics, the same equation is represented as \dot{A} = -i/h[A,H], where A and H are now quantum mechanical operators and [A,H] is the commutator, which also gives the rate of change of the observable A with respect to time.

Now, let's take a closer look at the given solution A(t) = e^{tL}A. This solution involves the exponential of the operator L, which is defined as Lf = (f, H)_{}, as mentioned in the forum post. This means that L is a linear operator that acts on functions of initial conditions (q,p) in the classical case or quantum mechanical operators in the Hilbert space.

To show that A(t) = e^{tL}A is the formal solution to \dot{A} = (A, H)_{}, we need to substitute this solution into the equation and show that it satisfies the equation. Let's do this step by step:

\dot{A} = (A, H)_{}

Substituting A(t) = e^{tL}A into the equation, we get:

\dot{A} = (\e^{tL}A, H)_{}

Using the linearity of the Poisson bracket, we can write this as:

\dot{A} = e^{tL}(A, H)_{}

Now, using the definition of L, we can write (A, H)_{} as L(A), which gives us:

\dot{A} = e^{tL}L(A)

Now, let's take a closer look at L(A). Using the definition of L, we can write L(A) as (A, H)_{}, which is the Poisson bracket. This means that