Time-Dependant Schro Composite System

[iamalexalright]

Homework Statement

Given:
$$a\partial\psi/\partial t = \widehat{H}\psi$$

Consider a is an unspecified constant. Show this equation has the following property. Let $$\widehat{H}$$ be the Hamiltonian of the system composed of two independent parts:
$$\widehat{H}(x_{1},x_{2}) = \widehat{H_{1}}(x_{1}) + \widehat{H_{2}}(x_{2})$$

and the stationary states of the composite system are:
$$\psi(x_{1},x_{2}) = \psi_{1}(x_{1},t)\psi_{2}(x_{2},t)$$

Show that this product form is a solution to the preceding equation for the given Hamiltonian.


2. The attempt at a solution
I have a feeling this is the wrong starting point but I do this (I assume I can use separation of variables)

$$\psi(x_{1},x_{2}) = \psi_{1}(x_{1},t)\psi_{2}(x_{2},t) = \vartheta_{1}(x_{1})T_{1}(t)\vartheta_{2}(x_{2})T_{2}(t)$$


After plugging that into the first equation and some algebra I separate the variables and then set them equal to some constant E. I then have:

$$\partial/\partial t (T_{1}T_{2}) = (1/a)ET_{1}T_{2}$$

and

$$\widehat{H}_{1}\vartheta_{1} + \widehat{H}_{2}\vartheta_{2} = E\vartheta_{1}\vartheta_{2}$$

and from here, assuming I did everything correctly, I don't know how to continue.

 

[kuruman]

** Edit ** Removed incorrect answer.

 

[kuruman]

I interpret

$$\widehat{H}(x_{1},x_{2}) = \widehat{H_{1}}(x_{1}) + \widehat{H_{2}}(x_{2})$$

to mean that

$$\widehat{H}(\vartheta_{1}\vartheta_{2}) = \vartheta_{2}\widehat{H_{1}}(\vartheta_{1}) + \vartheta_{1}\widehat{H_{2}}(\vartheta_{2})$$

Your interpretation is dimensionally incorrect, if nothing else. This should make a difference.