Suppose I have two normalised states $\Psi_1$ and $\Psi_2$ that are given to me as a sum of some orthonormal basis $\Psi_1 = \sum a_i \psi_i$, but suppose the sums are presented in some way that doesn't make it immediately obvious whether $\Psi_1$ and $\Psi_2$ are the same state.
If I calculate $< \Psi_1 | \Psi_2 >$ can I tell from this if the states are the same? Specifically if I get unity for the integral does that mean the two states are necessarily the same?
I am not sure if my thinking is correct and I'd like to ask if someone can confirm it, or give explanation, what am I doing wrong. I did task where I was asked to tell if pairs of expressions for quantum states represent the same state.
Two samples I've been thinking about:
$\frac{1}{\sqrt{2}}...
Would the answer be to simply calculate the $\langle \Psi_1 | \Psi_2 \rangle$ for each pair? Then if you get unity you know it's the same state while if the answer isn't unity they are different states.
Obviously in this case it's easy and you can tell the answer just by a casual look at the states, but if you wanted a formal procedure for testing would this be it?
@JohnRennie Yes, the formal statement is that equality in the Cauchy-Schwarz inequality $\lvert\langle \psi \vert \chi \rangle\rvert^2 \leq \langle \psi\vert\psi\rangle\langle\chi\vert\chi\rangle$ holds if and only if $\chi$ and $\psi$ are linearly dependent.
Well, string theory is not a quantum field theory, but you can construct the quantum field theory that reproduces the behaviour of the massless string states at "low" energies
Where "low" is...probably a rather gigantic energy, but I couldn't tell you any actual number.
the laziest way to see it (and probably oversimplified), is that creation operators should create allowed particle states, seen as vectors of the one-particle hilbert space
@HariPrasad : yes, and yes. And no, I haven't seen a magnetic monopole. You never will, because it's the electromagnetic field, so electric charge is a misnomer, so magnetic charge is a non-starter. Ask a question on the main site if you want more info.
Lets say we are working in a classical scalar field theory and we have two functional $ F[\phi, \pi](x)$ and $G[\phi, \pi](x)$. In most of the references, starting with two functional the Poisson bracket is defined as
$$\{F(x),G(y)\} = \int d^3z \left( \frac{\delta F(x)}{\delta \phi(z)}\frac{\del...
I am not sure if my thinking is correct and I'd like to ask if someone can confirm it, or give explanation, what am I doing wrong. I did task where I was asked to tell if pairs of expressions for quantum states represent the same state.
Two samples I've been thinking about:
$\frac{1}{\sqrt{2}}...
