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Say I have two states, one of which is obtained by transforming the other. If $|\psi_1\rangle$ is the 1st state, then the other state is expressed by: $|\phi_1\rangle=\hat{O}|\psi_1\rangle$ [$\hat{O}$ is the transformation]. So, $|\psi_1\rangle$ and $|\phi_1\rangle$ are related. Operating $\hat{O}$ on

\begin{equation} \mathcal{H_1}|\psi_1\rangle=E_1|\psi_1\rangle \end{equation} we see that:

\begin{align} \hat{O}\mathcal{H_1}|\psi_1\rangle &= \hat{O}E_1|\psi_1\rangle\\ \hat{O}\mathcal{H_1}\hat{O}^{-1}(\hat{O}|\psi_1\rangle)&=E_1(\hat{O}|\psi_1\rangle)\\ \implies\mathcal{H_2}|\phi_1\rangle&=E_1|\phi_1\rangle \end{align} where $\mathcal{H_2}=\hat{O}\mathcal{H_1}\hat{O}^{-1}$

My question is how to write the joint wave function of the total system. Is this meaningful to form the tensor products of $\lbrace|\psi_1\rangle\rbrace$ with $\lbrace|\phi_1\rangle\rbrace$? It is clear that $|\phi_1\rangle$ is not an eigenstate of $\mathcal{H_1}$. So, writing a general solution: $c_1|\psi_1\rangle+c_2|\phi_2\rangle$ does not seem to be possible.

Any light in this direction will be highly appreciated.

Regards, Kolahal

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    $\begingroup$ "My question is how to write the joint wave function of the total system." What is the system? You haven't specified what the system is, or what the different degrees of freedom are, etc. You only use a tensor product when you have states in different Hilbert spaces corresponding to different degrees of freedom. Here, you have two states in the same Hilbert space related by a (presumably unitary?) transformation. What exactly are you trying to do? $\endgroup$ – march Dec 29 '17 at 22:43
  • $\begingroup$ I am trying to frame a quantum mechanical model of image charge obtained in the grounded conducting sphere image problem. The states $|\psi\rangle$ and $|\phi\rangle$ are inverted w.r.t. a sphere. If $|\psi\rangle$ represents a real charge, the transformed wave function $|\phi\rangle$ will represent the image. In fact, this transformation is non-unitary (however, canonical). Due to this, the form of $\mathcal{H_1}$ is different from that of $\mathcal{H_2}$ and I cannot write a general solution as a simple linear combination of these two wave packets. I am trying to write the general solution. $\endgroup$ – kolahalb Dec 30 '17 at 0:13
  • $\begingroup$ I think more details are required. What are the degrees of freedom that you are quantizing? Are they field variables? In that case, you need to go a QFT route and quantize the electromagnetic modes matching the boundary conditions at the conducting sphere (I'm not sure how to do this!). Is there an atom feeling this field, and you can treat the field classically? In that case, the atom just feels the local electric field, which you can solve for classically. The point is, in order to help, I think we need a lot more detail about the physical setup of the problem. $\endgroup$ – march Dec 30 '17 at 0:33
  • $\begingroup$ We have just standard canonical quantization of position and momentum operators. The field is classical. It might be helpful to imagine a wave function in front of the sphere. I can calculate image wave function. The boundary condition is that: these wave functions vanish on the sphere. There should be a way to get the total wave function. If field is a concern, I would like to know what happens if there is no field at all. Suppose I am talking only about inverted wave function with no EM interaction. That is: $V=0$ at $r\ge R_0$ and $V\rightarrow\infty$ at $r\le R_0$. $\endgroup$ – kolahalb Dec 30 '17 at 1:46
  • $\begingroup$ Im sorry if I’m being obtuse, but position and momentum of what, and how does that relate to the electric field? $\endgroup$ – march Dec 30 '17 at 1:56

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