Probability of getting specific states -- Quantum Mechanics

[Thomas Brady]

I'm pretty new to quantum, so I'm pretty sure I'm missing something basic here. I've got a 4x4 Hamiltonian with eigenkets $$\psi_{U} = 1/(\sqrt 2) (\psi_{1up} \pm \psi_{2up})$$ and $$\psi_{D} = 1/(\sqrt 2) (\psi_{1down} \pm \psi_{2down})$$ The only difference between the two states is the spin as signified by the subscripts up (U) and down (D). The plus states have the eigenvalue $E_0 - t$ and the minus states have the eigenvalue $E_0 + t$. Knowing this, how can I say what the probability is of an electron being in any of the $\psi_{1up}$, $\psi_{1down}$, $\psi_{2up}$, and $\psi_{2down}$ states, without knowing the probability it is in $\psi_{U}$ or $\psi_{D}$?

Just to be clear there are of course 4 eigenkets, but the difference between each of the plus and minus eigenkets is the spin

 

[blue_leaf77]

Knowing this, how can I say what the probability is of an electron being in any of the ψ1up\psi_{1up}, ψ1down\psi_{1down}, ψ2up\psi_{2up}, and ψ2down\psi_{2down} states, without knowing the probability it is in ψU\psi_{U} or ψD\psi_{D}?

The basis sets $\psi_u^+,\psi_u^-,\psi_d^+,\psi_u^-$ and $\psi_{1\textrm {up}},\psi_{1\textrm {down}},\psi_{2\textrm {up}},\psi_{2\textrm {down}}$ are connected by some transformation matrix. Thus, a given state of an electron can be equivalently expressed in either set. If you don't know the expansion coefficients, which represents the probability being found in the corresponding basis, in one of the sets, you cannot determine the coefficients in the other set.

 

[Thomas Brady]

The basis sets $\psi_u^+,\psi_u^-,\psi_d^+,\psi_u^-$ and $\psi_{1\textrm {up}},\psi_{1\textrm {down}},\psi_{2\textrm {up}},\psi_{2\textrm {down}}$ are connected by some transformation matrix. Thus, a given state of an electron can be equivalently expressed in either set. If you don't know the expansion coefficients, which represents the probability being found in the corresponding basis, in one of the sets, you cannot determine the coefficients in the other set.

So, essentially I cannot determine the probabilities, with the given information

 

[blue_leaf77]

with the given information

The given information? You mean the equations you posted there? Those vectors you gave there are merely the eigenvectors of the Hamiltonian. An electron can have arbitrary wavefunction which is some linear combination of those 4 vectors. In order to know the probabilities you have to know the state the electron is in.

 

[Thomas Brady]

The given information? You mean the equations you posted there? Those vectors you gave there are merely the eigenvectors of the Hamiltonian. An electron can have arbitrary wavefunction which is some linear combination of those 4 vectors. In order to know the probabilities you have to know the state the electron is in.

Right, and with the problem I was given it does not appear to specify which state it is in.

 

[blue_leaf77]

Right, and with the problem I was given it does not appear to specify which state it is in.

Yes, it doesn't give you the information about the state.