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

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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.
  • #1
benfrombelow
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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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  • #2
What does it mean that an operator is conserved ?

Daniel.
 
  • #3
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
 
  • #4
And what does "conserved" mean for an antilinear operator ?

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

Daniel.
 
  • #5
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...
 
  • #6
I don't seem to get the line of thought in your calculations. Could you care to explain how did you get from

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

to

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

Daniel.
 
  • #7
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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  • #8
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

[tex] \langle \psi (t), \psi(t) \rangle \neq \langle \psi (0), \psi (0) \rangle [/tex]

Daniel.
 

Related to Prove T Not Conserved When [H,T] = 0

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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