- #1
Nomajere
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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!
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!
