Group Theory why transformations of Hamiltonian are unitary?

In summary, the conversation is discussing the matrix D and its role in transforming the wave function, specifically in preserving some property. The person speaking is inquiring about the necessity for D to be unitary, and the other person clarifies that it is only required if the goal is to preserve vector lengths. The conversation also mentions that in lectures, D has been shown to be a representation of T-I, and the person is trying to understand the proof. The other person suggests that this is only true if the position vector r is scaled.
  • #1
applestrudle
64
0
This is what I have so far:

part1.png
part2.png


I'm trying to show that the matrix D has to be unitary. It is the matrix that transforms the wavefunction.
 
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  • #2
The matrix that transforms the wave function how? So that it preserves some property? Transformations do NOT have to be unitary unless you are trying to lengths of vectors.
 
  • #3
HallsofIvy said:
The matrix that transforms the wave function how? So that it preserves some property? Transformations do NOT have to be unitary unless you are trying to lengths of vectors.

In lectures we were showing
Tψ(r) = ψ(Ur) = ΣDij ψ(r)

Dij has to be unitary and form a representation of T - I'm just trying to figure out the proof. Are you saying this is only try if you scale the position vector r?
 

Related to Group Theory why transformations of Hamiltonian are unitary?

1. What is Group Theory?

Group Theory is a branch of mathematics that deals with the study of groups, which are sets of elements that follow a defined set of rules for combining or transforming them.

2. How is Group Theory applied in science?

Group Theory has various applications in science, including physics, chemistry, and computer science. In physics, it is used to understand the symmetries of physical systems and describe the behavior of particles. In chemistry, it is used to understand the electronic structure of molecules and predict their properties. In computer science, it is used to study algorithms and data structures.

3. What is a Hamiltonian?

A Hamiltonian is a mathematical operator that represents the total energy of a system in quantum mechanics. It is used to describe the dynamics of a physical system and predict its future behavior.

4. Why are transformations of Hamiltonian important?

Transformations of Hamiltonian are important because they allow us to describe the evolution of a quantum system over time. By applying these transformations, we can understand how a system changes under different conditions and predict its behavior.

5. Why are transformations of Hamiltonian unitary?

Transformations of Hamiltonian are unitary because they preserve the normalization of the state vector, which is a fundamental property in quantum mechanics. This means that the total probability of finding a particle in any state is always equal to 1, regardless of the transformation applied to the system.

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