Question concerning Hamiltonian and eigenstates

[Gordijnman]

Homework Statement

Two spin-1/2 particles are placed in a system described by Hamiltonian H=S(x1)S(x2), (S(x) being the spin operator in the x-direction). States are written like |$$\uparrow\downarrow$$>, (and can be represented by 2 x 2 matrix) so that there are 4 possible states. (|$$\uparrow\uparrow>, |\uparrow\downarrow>, |\downarrow\downarrow>, |\downarrow\uparrow>$$

Given: |$$\phi> = |\uparrow\downarrow> - |\downarrow\uparrow>$$

Find the normalized eigenstate of |$$\phi$$>

Homework Equations

S(x1) and S(x2) are both matrices represented by $$\hbar$$/2 *
0 1 (I don't know how to write matrices), where x1 operates on the first particle, and x2 on the second.
1 0

The Attempt at a Solution

S(x) inverts the spin, and multiplies the state by $$\hbar/2$$; there are 2 spin operators working on the different states so that the whole state in the end will be multiplied by the square of that: $$\hbar\stackrel{2}{}/4$$.
The difficulty is that I don't understand what is meant by normalized eigenstate. I do end up with a eigenstate (or energy) of -$$\hbar\stackrel{2}{}/4$$ by calculating <$$\phi|\phi>$$, but I'm quite sure that's not the definite answer, because something still needs to be normalized.

 

[mark726]

Thank you for your post. I can help you understand what is meant by a normalized eigenstate and how to find it in this specific situation.

First, let's define what an eigenstate is. An eigenstate is a state of a system that, when operated on by a specific operator, returns the same state multiplied by a constant factor. In this case, the operator is the Hamiltonian, and the eigenstates are the possible states of the system.

Now, what does it mean for a state to be normalized? A normalized state is one that has a probability of 1 of being measured in that state. In other words, the sum of the probabilities of all possible states must equal 1. In this case, the states are the possible combinations of spin orientations for the two particles.

To find the normalized eigenstate of |\phi>, we need to first find the eigenstates of the system. These are the states that, when operated on by the Hamiltonian, return the same state multiplied by a constant factor. In this case, the eigenstates are |\uparrow\uparrow>, |\uparrow\downarrow>, |\downarrow\downarrow>, and |\downarrow\uparrow>.

Next, we need to find the constant factor for each eigenstate. This can be done by operating on each eigenstate with the Hamiltonian and setting it equal to the original state multiplied by the constant factor. For example, for the state |\uparrow\downarrow>, we have:

H|\uparrow\downarrow> = \hbar\stackrel{2}{}/4 * |\uparrow\downarrow>

S(x1)S(x2)|\uparrow\downarrow> = \hbar\stackrel{2}{}/4 * |\uparrow\downarrow>

(-\hbar\stackrel{2}{}/4)|\uparrow\downarrow> = \hbar\stackrel{2}{}/4 * |\uparrow\downarrow>

Therefore, the constant factor for the state |\uparrow\downarrow> is -1.

Following the same process for the other eigenstates, we get the following constant factors: 1 for |\uparrow\uparrow>, -1 for |\uparrow\downarrow>, -1 for |\downarrow\downarrow>, and 1 for |\downarrow\uparrow>.

Now, to find the normalized eigenstate of |\