# Prove T Not Conserved When [H,T] = 0

• benfrombelow
In summary, the conversation discusses the concept of conservation for an antiunitary operator, specifically in relation to the Hamiltonian. It is stated that if [H,A]=0, then the operator is considered conserved in the Heisenberg picture. However, this is not true for antiunitary operators. The conversation then delves into a mathematical proof, using the Dirac's bra-ket formalism, to show that T is not conserved in an arbitrary state. The improper use of the formalism and the lack of self-adjointness of T result in an invalid equality.
benfrombelow
If [H,T] = 0 (where H is the Hamiltonian and T is the (antiunitary) time-reversal operator) prove that T is NOT conserved

This is for a homework assignment due is less than 12 hours...

I'm guessing it has to with the property of antiunitary operators: T|a> = |a'> then <a|b> = <a'|b'>* but it's late and I'm lazy

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What does it mean that an operator is conserved ?

Daniel.

Usually, it means that the operator is constant, in the Heisenberg picture, which means that [H,A]=0 but that isn't true when A is anti-linear

And what does "conserved" mean for an antilinear operator ?

If you don't know that, you can't solve it.

Daniel.

The best I can come up with is to look at the expectation value of T in an arbitrary state: <Psi(t)|T|Psi(t)> = <Psi(0)|(exp(iHt/h))T(exp(-iHt/h))|Psi(0)> = <Psi(0)|(exp(2*iHt/h))T|Psi(0)> = <Psi(-2t)|T|Psi(0)> = <Psi(0)|T|Psi(2t)>

which clearly isn't constant, but it also isn't very satisfying...

I don't seem to get the line of thought in your calculations. Could you care to explain how did you get from

$$\langle \Psi(0)|\mbox{(exp(iHt/h))}T\mbox{(exp(-iHt/h))}|\Psi(0)\rangle$$

to

$$\langle \Psi(0)|\mbox{(exp(2*iHt/h))}T|\Psi(0) \rangle$$

Daniel.

Well, basically, since T is antilinear, TiH = -iTH. But T commutes with H, so TiH = -iHT.

edit: To elaborate, because TiH = -iHT, it follows that T(exp(-iHt/h)) = (exp(iHt/h))T since the exponential is a series in powers of H. When moving T to the right of H, the coefficient of each term in the series must be replaced with its complex conjugate

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Yes, i can see now. One thing to notice is the improper use of Dirac's bra-ket formalism for T, since this operator is not self adjoint.
Thus the last equality does not follow. You managed to find (neglecting the last equality) that

$$\langle \psi (t), \psi(t) \rangle \neq \langle \psi (0), \psi (0) \rangle$$

Daniel.

## 1. What does it mean for T to be conserved?

Conservation of a quantity means that its value remains constant over time, even as other variables may change. In physics, conservation laws are fundamental principles that describe the behavior of physical systems.

## 2. What is the significance of [H,T] = 0 in regards to T not being conserved?

The expression [H,T] = 0 means that the Hamiltonian (H) and observable (T) do not commute, or their operators do not yield the same result when applied in either order. This indicates that T is not a conserved quantity in the system described by H.

## 3. Can you provide an example of a physical system where T is not conserved when [H,T] = 0?

One example is a particle moving in a potential well. The Hamiltonian in this system describes the energy of the particle, while the observable T represents its position. Since the particle's position changes over time, T is not conserved and [H,T] = 0.

## 4. How does this concept relate to the Heisenberg uncertainty principle?

The Heisenberg uncertainty principle states that the more precisely we know a particle's position, the less precisely we can know its momentum, and vice versa. This uncertainty in measuring both quantities simultaneously is related to the non-commutativity of their operators, which is reflected in the fact that [H,T] = 0 when T is not conserved.

## 5. Is the non-conservation of T always a consequence of [H,T] = 0?

No, there are cases where T is not conserved even when [H,T] ≠ 0. This can occur in systems with time-varying Hamiltonians or when T is not a well-defined quantity. In these cases, other methods such as Noether's theorem may be used to determine if T is conserved or not.

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