# Quantum Mechanics and Group Theory questions

• Nomajere
In summary: I feel like it's too abstract for me, but I am still trying to learn. In summary, I think Griffiths is great for conceptual understanding but may not be so good when it comes to getting the details down.
Nomajere
Hello all. I am new here. I am in the last quarter of a 3 quarter sequence of undergrad quantum mechanics and I just had some conceptual questions (nothing pertaining to homework). We just recently covered Berry's Phase and the Dynamical Phase. Now I wanted to start with a more basic quantum mechanical question before moving to Berry's Phase:

When determining a wavefunction $$\psi$$ we can pretty much derive a wavefunction "up to" a phase, i.e $$\psi = A \left(functions\right) e^{i \chi}$$ Now, if I am understanding elementary group theory correctly, a U(1) transformation would be multiplying our initial function by a complex exponential with a phase. So, is the wavefunction I wrote "up to a phase" basically transformed by U(1)?

Okay, assuming what I first asked is legitimate, I wanted to ask about Berry and Dynamical Phases. Through the adiabatic approximation we can write $$\Psi_n\left(t\right) = \psi_n \left(t \right) e^{i \theta_n \left(t \right)} e^{i \gamma_n \left(t \right)}$$. My question here is if any of this may be similar or is a U(1) transformation, or is it not so because of the possible time dependence?

Relevant equations:
$$\theta_n \left(t \right) =-\frac{1}{\hbar} \int_{0}^{t} E_n\left(t'\right) dt'$$

$$\gamma_{n} \left(t \right) = i \int_{0}^{t} \langle \psi_m\left(t'\right) | \frac{\partial}{\partial t'} \psi_{m}\left(t' \right) \rangle dt'$$

Thank you all for your patience and help!

I'm not sure I'm understanding your two questions but I will try to do the best with my (not that extensive) knowledge of the matter

- Yes, multiplying by a phase is equivalent to doing a U(1) transformation.

- if the adiabatic approximation is valid and If the path is closed, meaning H(T)=H(t=0), then yes the acquired geometrical phase is a very specific U(1) transformation linking the psi(t=0) original vector to the psi(T) final vector. It is specific in the sense that it depend on the path traveled by the hamiltonian (or more generally the path traveled in the projective hilbert space).

andresB said:
I'm not sure I'm understanding your two questions but I will try to do the best with my (not that extensive) knowledge of the matter

- Yes, multiplying by a phase is equivalent to doing a U(1) transformation.

- if the adiabatic approximation is valid and If the path is closed, meaning H(T)=H(t=0), then yes the acquired geometrical phase is a very specific U(1) transformation linking the psi(t=0) original vector to the psi(T) final vector. It is specific in the sense that it depend on the path traveled by the hamiltonian (or more generally the path traveled in the projective hilbert space).

Thank you! That was basically what I was looking for. We are using Griffiths, which when it comes to showing me how to do quantum problems, is great. Conceptually, I am not too happy with it all the time.

## 1. What is the relationship between quantum mechanics and group theory?

Quantum mechanics and group theory are both branches of theoretical physics that overlap in the study of symmetries and transformations. Group theory provides a mathematical framework for understanding symmetries in quantum systems, while quantum mechanics uses group theory to describe and predict the behavior of particles on a microscopic level.

## 2. Why is group theory important in quantum mechanics?

Group theory is important in quantum mechanics because it helps us understand the symmetries and transformations that govern the behavior of particles. It provides a powerful tool for analyzing the properties of quantum systems and predicting their behavior. Additionally, group theory allows us to classify and organize particles based on their symmetries, which helps us make sense of the complex world of quantum mechanics.

## 3. How are Lie groups and Lie algebras used in quantum mechanics?

Lie groups and Lie algebras are mathematical structures that are used to describe continuous symmetries in quantum systems. They provide a way to represent and manipulate the symmetries of a quantum system, which is crucial for understanding its properties and predicting its behavior. In quantum mechanics, Lie groups and Lie algebras are used to study a wide range of physical phenomena, from the behavior of subatomic particles to the properties of materials.

## 4. What is the significance of symmetry in quantum mechanics?

Symmetry plays a crucial role in quantum mechanics because it helps us understand the behavior of particles and systems. In quantum mechanics, symmetries can reveal hidden relationships between different states or particles, and can even lead to the discovery of new particles or properties. Additionally, symmetries can be used to simplify complex systems and make predictions about their behavior.

## 5. How does group theory relate to practical applications in quantum technologies?

Group theory has many practical applications in quantum technologies, such as quantum computing and quantum cryptography. It provides a powerful tool for analyzing the behavior of quantum systems and designing new technologies that take advantage of quantum properties. For example, the symmetries of quantum systems can be used to encode and manipulate information in quantum computers, leading to more efficient and powerful computing capabilities.

Replies
5
Views
800
Replies
2
Views
2K
Replies
5
Views
2K
Replies
27
Views
2K
Replies
5
Views
1K
Replies
22
Views
2K
Replies
8
Views
2K
Replies
7
Views
959
Replies
9
Views
1K
Replies
8
Views
1K