# If Aop commutes with H, A is a constant of motion - prove

• lavster
In summary, the conversation discusses the statement that <A> = (1/ih)[A_{op}, H] if A_{op} is independent of time. The participants also mention the Hamiltonian, expectation value, and commutator, and how they relate to the above equation. They also discuss the importance of A being a constant of motion and how it can be a good quantum number. The conversation ends with a suggestion to observe that the equation implies the expectation value of the commutator.

#### lavster

hi i was reading a textbook and this statement puzzled me. it stated that $$\frac{d}{dt}<A>=\frac{1}{ih}[A_{op},H] if \frac{\partial A_{op}}{ \partial t}=0$$.

i was wanting to prove this and hence show that if Aop commutes A is a constant of motion and can be a good quantum number.

I get that: H is the hamiltonian expressed as follows: $$(H\phi)=i\hbar\frac{\partial \phi}{\partial t}$$ , <A> is the expectation value: $$<A>=\int{\phi^*A_{op}\phi d\tau}$$ and [A_{op},H] is the commutator $$(A_{op}H-HA_{op})$$ and this equals zero when it commutes. However i can't put it together to get the above equation. can someone show me how to do it please?

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Would it help if I observed that it's implicit that it really means the expectation value of the commutator ?

lavster said:
<A> is the expectation value: $$<A>=\int{\phi^*A_{op}\phi d\tau}$$
$d\tau$? It should be $d^3x$, and things get easier if you write down how these things depend on time.

$$\langle A\rangle_{\phi(t)}=\langle\phi(t),A\phi(t)\rangle=\langle e^{-iHt}\phi,Ae^{-iHt}\phi\rangle=\langle\phi,e^{iHt}Ae^{-iHt}\phi\rangle$$

I'm using units such that $\hbar=1$.

## 1. What does "Aop commutes with H" mean?

"Aop commutes with H" means that the operator Aop and the Hamiltonian operator H can be rearranged in any order without changing the outcome of the equation.

## 2. How does this relate to the concept of constants of motion?

If Aop commutes with H, it means that Aop is a constant of motion. This means that the quantity represented by Aop remains unchanged over time, even as other variables in the system may change.

## 3. What is the significance of A being a constant of motion?

The fact that A is a constant of motion tells us that there is an underlying symmetry in the system. This symmetry can provide valuable insights into the behavior of the system and can help us make predictions about its future state.

## 4. How can you prove that A is a constant of motion if it commutes with H?

To prove that A is a constant of motion, we must show that it satisfies the condition of being conserved over time. This can be done by showing that the commutator of A with the Hamiltonian is equal to zero, which indicates that A does not change over time.

## 5. Can you give an example of a physical system where A is a constant of motion?

One example is a pendulum. In this system, the Hamiltonian operator represents the total energy of the pendulum, and the angular momentum operator commutes with it. This means that angular momentum is conserved and therefore a constant of motion in this system.