- #1
Hazzattack
- 69
- 1
Hi everyone, I was hoping that someone might be able to tell me if what I'm doing is legit.
Firstly, I start by saying that a unitary transform can be made between two sets of operators (this is defined in this specific way);
b[itex]_{n}[/itex] = [itex]\Sigma_{m}[/itex] U[itex]_{mn}[/itex]a[itex]_{m}[/itex] (1)
Now this is the bit I'm not sure about, if i want to do the inverse to solve for a[itex]_{m}[/itex] is the following acceptable;
[itex]\Sigma_{n}[/itex]U[itex]_{nm}[/itex]^{dagger}b[itex]_{n}[/itex] = [itex]\Sigma_{n}[/itex][itex]\Sigma_{m}[/itex]U[itex]_{mn}[/itex]U[itex]_{nm}[/itex]^{dagger}a[itex]_{m}[/itex]
Where [itex]\Sigma_{n}[/itex][itex]\Sigma_{m}[/itex]U[itex]_{mn}[/itex]U[itex]_{nm}[/itex]^{dagger} = (UU^{dagger})[itex]_{nn}[/itex]= [itex]\delta_{nn}[/itex] = 1
Thanks in advance.
If this is entirely wrong, some pointers on how I isolate a[itex]_{m}[/itex] would be appreciated.
In essence what I'm asking is given (1), what is the inverse transformation of it?
Extra information: The components of U[itex]_{mn}[/itex] are real.
Firstly, I start by saying that a unitary transform can be made between two sets of operators (this is defined in this specific way);
b[itex]_{n}[/itex] = [itex]\Sigma_{m}[/itex] U[itex]_{mn}[/itex]a[itex]_{m}[/itex] (1)
Now this is the bit I'm not sure about, if i want to do the inverse to solve for a[itex]_{m}[/itex] is the following acceptable;
[itex]\Sigma_{n}[/itex]U[itex]_{nm}[/itex]^{dagger}b[itex]_{n}[/itex] = [itex]\Sigma_{n}[/itex][itex]\Sigma_{m}[/itex]U[itex]_{mn}[/itex]U[itex]_{nm}[/itex]^{dagger}a[itex]_{m}[/itex]
Where [itex]\Sigma_{n}[/itex][itex]\Sigma_{m}[/itex]U[itex]_{mn}[/itex]U[itex]_{nm}[/itex]^{dagger} = (UU^{dagger})[itex]_{nn}[/itex]= [itex]\delta_{nn}[/itex] = 1
Thanks in advance.
If this is entirely wrong, some pointers on how I isolate a[itex]_{m}[/itex] would be appreciated.
In essence what I'm asking is given (1), what is the inverse transformation of it?
Extra information: The components of U[itex]_{mn}[/itex] are real.
Last edited: