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- is spin up or spin down higher in energy?
I am confused about why spin down has a lower energy than spin up. What is the correct interpretation of the equations?
If we have a spin ##\frac{1}{2}## particle in a magnetic field ##B_0## that is applied in the positive z direction
The spin states of the particle are
$$\ket{up} = \begin{bmatrix} 1\\0\end{bmatrix}$$
$$\ket{down} = \begin{bmatrix} 0\\1\end{bmatrix}$$
and the hamiltonian for the spin in the magnetic field is $$\hat{H}=\frac{\hbar\omega}{2}\sigma_z$$
to find the energies of the spin states, we operate on ##\ket{up}## and ##\ket{down}## with ##\hat{H}##
$$\hat{H}\ket{up} =\frac{\hbar\omega}{2} \begin{bmatrix} 1 & 0\\0 & -1\end{bmatrix}\begin{bmatrix} 1\\0\end{bmatrix}=\frac{\hbar\omega}{2}\begin{bmatrix} 1\\0\end{bmatrix}=\frac{\hbar\omega}{2}\ket{up}$$
$$\hat{H}\ket{down} =\frac{\hbar\omega}{2} \begin{bmatrix} 1 & 0\\0 & -1\end{bmatrix}\begin{bmatrix} 0\\1\end{bmatrix}=-\frac{\hbar\omega}{2}\begin{bmatrix} 1\\0\end{bmatrix}=-\frac{\hbar\omega}{2}\ket{down}$$
so the energy of ##\ket{up}## is ##\frac{\hbar\omega}{2}## and the energy of ##\ket{down}## is ##-\frac{\hbar\omega}{2}##. The energy of ##\ket{down}## is lower than up..and it is a negative value. are states allowed to have negative energies? why does down have a lower energy than up?
If we have a spin ##\frac{1}{2}## particle in a magnetic field ##B_0## that is applied in the positive z direction
The spin states of the particle are
$$\ket{up} = \begin{bmatrix} 1\\0\end{bmatrix}$$
$$\ket{down} = \begin{bmatrix} 0\\1\end{bmatrix}$$
and the hamiltonian for the spin in the magnetic field is $$\hat{H}=\frac{\hbar\omega}{2}\sigma_z$$
to find the energies of the spin states, we operate on ##\ket{up}## and ##\ket{down}## with ##\hat{H}##
$$\hat{H}\ket{up} =\frac{\hbar\omega}{2} \begin{bmatrix} 1 & 0\\0 & -1\end{bmatrix}\begin{bmatrix} 1\\0\end{bmatrix}=\frac{\hbar\omega}{2}\begin{bmatrix} 1\\0\end{bmatrix}=\frac{\hbar\omega}{2}\ket{up}$$
$$\hat{H}\ket{down} =\frac{\hbar\omega}{2} \begin{bmatrix} 1 & 0\\0 & -1\end{bmatrix}\begin{bmatrix} 0\\1\end{bmatrix}=-\frac{\hbar\omega}{2}\begin{bmatrix} 1\\0\end{bmatrix}=-\frac{\hbar\omega}{2}\ket{down}$$
so the energy of ##\ket{up}## is ##\frac{\hbar\omega}{2}## and the energy of ##\ket{down}## is ##-\frac{\hbar\omega}{2}##. The energy of ##\ket{down}## is lower than up..and it is a negative value. are states allowed to have negative energies? why does down have a lower energy than up?