# Time derivative in the Heisenberg picture?

• pellman
In summary, the difference between total and partial derivatives is that partial derivatives are time-dependent expressions of the change in a function over time, while total derivatives are expressions of the change in a function over space.
pellman

$$\frac{d}{dt}A(t)=\frac{i}{\hbar}[H,A(t)]+\left(\frac{\partial A}{\partial t}\right)$$

I don't understand what the distinction between

$$\frac{d}{dt}A(t)$$ and $$\left(\frac{\partial A}{\partial t}\right)$$

is supposed to be. That is, what is the difference between the meaning of these two expressions?

For regular old c-number functions, the difference between total and partial derivatives is something like

$$\frac{df}{dt}=\frac{\partial f}{\partial u}\frac{du}{dt}+\frac{\partial f}{\partial t}$$.

where f = f(u,t). If f doesn't depend on other variables, then $$\frac{df}{dt}=\frac{\partial f}{\partial t}$$.

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Recall that in the Heisenberg Picture we express the time evolution of the observables for a dynamically evolving system by time varying the operators, leaving the Hilbert space vectors (e.g. wave-functions) fixed at their initial values.

So for a fixed observable, there will be a time dependence relating to the evolution of the system being observed. That is the component of the total time derivative expressed by the commutator.

But in the general setting we can also consider a time varying observable i.e. imagine a polarizer which is rotating over time. This explicit time evolution of the observational device is expressed by an explicit time dependence of the operator and that's the partial derivative w.r.t. t component.

Again this is somewhat confusing since in Physics we too often express both function and functional value (variable) with the same symbol.

Imagine the observable A(t) in the Heisenberg picture is expressed as a function 'a' of the initial operator (the operator corresponding to measuring the system at time t=0) and also a function of t directly indicating a time evolution of the measuring device:
$$A(t) = a(t,A_0)$$.

Then
$$\frac{dA}{dt} = \frac{i}{\hbar} [H,A(t)] + \frac{\partial a}{\partial t}$$

It may be helpful to think of the A as the correspondent of the value of a classical observable. Then consider say a position as seen by a moving observer relative to a given frame. There are two components, the motion of the particle (in the given frame) and the motion of the observer. With frame transformation x' = x+ X(t) we have:
dx'/dt = dx/dt + dX/dt which becomes in QM [H,x] + dX/dt.

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That was very helpful, jambaugh. thank you!

## 1. What is the Heisenberg picture in quantum mechanics?

The Heisenberg picture is one of two formalisms used in quantum mechanics, along with the Schrödinger picture. In this picture, the operators representing observables such as position and momentum are time-independent, while the state vector evolves in time. This is in contrast to the Schrödinger picture, where the operators are time-dependent and the state vector is time-independent.

## 2. How is the time derivative defined in the Heisenberg picture?

In the Heisenberg picture, the time derivative of an operator A is defined as:

dA/dt = (1/iℏ)[H, A]

where H is the Hamiltonian operator and ℏ is the reduced Planck's constant. This equation is known as the Heisenberg equation of motion.

## 3. What is the physical significance of the time derivative in the Heisenberg picture?

The time derivative in the Heisenberg picture represents the rate of change of an operator with respect to time. This is important in quantum mechanics because it allows us to study the evolution of observables over time, and how they are affected by the Hamiltonian of the system.

## 4. How does the time derivative in the Heisenberg picture relate to the Schrödinger picture?

In the Schrödinger picture, the time derivative of an operator is given by:

dA/dt = (1/iℏ)[H(t), A]

where H(t) is the time-dependent Hamiltonian. This means that the time derivative in the Schrödinger picture is related to the time derivative in the Heisenberg picture by a time-dependent unitary transformation.

## 5. What are the advantages of using the Heisenberg picture over the Schrödinger picture?

One advantage of using the Heisenberg picture is that it simplifies the equations of motion for operators, as they are time-independent. This can make calculations easier and more intuitive. Additionally, the Heisenberg picture is often used in systems with a time-independent Hamiltonian, as it allows for a more straightforward analysis of the system's dynamics.

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