The generator of time translation

In summary, the Hamiltonian is often referred to as the generator of time translation in quantum mechanics because it is related to the exponential function, which is used to represent the time evolution of quantum systems. This concept is also seen in other operators, such as the momentum operator and angular momentum operator, which generate translations and rotations, respectively. Despite not always being explicitly discussed in books on quantum mechanics, this perspective is rooted in the theory of Lie groups and algebras.
  • #1
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It is not hard to show that for QM what takes the wave function from t->t+Δt is the exponential of the hamiltonian. Yet for some mysterious reason my book decides to note the Hamiltonian as the generator of time translation rather than the exponential of it. What is the reason for this?
 
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  • #2
In the theory of lie groups a generator is defined as the element of the tangential lie algebra that generates the group element. The map from the lie algebra to the lie group is the exponential, so the generator of time evolution is the hamiltonian and the one dimensional subgroup it generates is the time evolution exp(i H t).

You find the same situation for the momentum operator and the spatial translation group it generates. And angular momentum operators generate rotations.
 
  • #3
Books on QM tend to speak from the perspective of group theory, even though they don't/may not contain chapters on Lie groups/algebras and their applications to quantum physics.
 
  • #4
The lie algebra has to do with what happens close to the identity. We can take what is given: [itex]e^{-i\hat{H}t}[/itex] is the generator of finite time translations, and see how this portrays the Hamiltonian.
[tex]
e^{-i\hat{H}\epsilon / \hbar}\psi(x,t) = \psi(x,t+\epsilon)
[/tex]
now for [itex]\epsilon \approx 0[/itex] we can write
[tex]
\psi(x,t)-\frac{i\hat{H}\epsilon}{\hbar}\psi(x,t) +\mathcal{O}(\epsilon^2) = \psi(x,t)+ \left. \frac{\partial \psi(x,t)}{\partial t}\right|_{t=\epsilon}\epsilon +\mathcal{O}(\epsilon^2)
[/tex]
thus
[tex]
\hat{H} = i\hbar \frac{\partial}{\partial t}
[/tex]
 

FAQ: The generator of time translation

1. What is the generator of time translation?

The generator of time translation is a mathematical concept in physics that describes the transformation of time in a physical system. It is a fundamental concept in the theory of relativity and is used to calculate the effects of time dilation and time reversal.

2. How is the generator of time translation calculated?

The generator of time translation is calculated by taking the time derivative of the Hamiltonian of a system. This is represented by the symbol Ḣ, and it measures the rate of change of the Hamiltonian with respect to time.

3. What is the significance of the generator of time translation?

The generator of time translation is significant because it is a fundamental quantity in the theory of relativity and is used to describe the behavior of time in physical systems. It allows us to calculate the effects of time dilation and time reversal, which have significant implications in fields such as cosmology and particle physics.

4. How does the generator of time translation relate to the concept of space-time?

The generator of time translation is closely related to the concept of space-time, which is a mathematical framework that combines space and time into a single entity. The generator of time translation is one of the four generators of the symmetry group of space-time, along with generators for spatial translations, rotations, and boosts.

5. What are some real-world applications of the generator of time translation?

The generator of time translation has numerous real-world applications, including in the fields of astrophysics, particle physics, and engineering. It is used to calculate the effects of time dilation in GPS satellites, to study the behavior of particles in high-energy accelerators, and to understand the evolution of the universe. It also has practical applications in designing accurate clocks and timekeeping devices.

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