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For a Hilbert space of two dimensions and hence two energy state eigenvectors, $\omega=\frac{E_2-E_1}{\hbar}$ is the frequency of probability oscillation(?) of being in the original state, where the operator is time independent and does not commute with the Hamiltonian
Significance
This is very important in MRI.Derivation
Let $\ket{a}=a_1\let{E_1}+a_2\ket{E_2}$. Then by the solution to the time independent Shrodinger's equation, $$\begin{align*} \ket{\psi(t)}=c_1e^{-iE_1t/\hbar}\ket{E_1}+c_2^{-iE_2t\/hbar}\ket{E_2}. \end{align*}$$We use the identity that for a complex number $z=a+ib$, $\abs{z}^2=|a+b|^2=|a|^2+|b|^2+2\text{Re}(a*b)$. Then the proability of being in the state $\ket{a}$ is $$\begin{align*} \abs{\braket{a|\psi(t)}}^2 \end{align*}$$
Associated Concepts
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For a Hilbert space of two dimensions and hence two energy state eigenvectors, $\omega=\frac{E_2-E_1}{\hbar}$ is the frequency of probability oscillation(?) of being in the original state, where the operator is time independent and does not commute with the Hamiltonian
Significance
This is very important in MRI.Derivation
Let $\ket{a}=a_1\let{E_1}+a_2\ket{E_2}$. Then by the solution to the time independent Shrodinger's equation, $$\begin{align*} \ket{\psi(t)}=c_1e^{-iE_1t/\hbar}\ket{E_1}+c_2^{-iE_2t\/hbar}\ket{E_2}. \end{align*}$$We use the identity that for a complex number $z=a+ib$, $\abs{z}^2=|a+b|^2=|a|^2+|b|^2+2\text{Re}(a*b)$. Then the proability of being in the state $\ket{a}$ is $$\begin{align*} \abs{\braket{a|\psi(t)}}^2 \end{align*}$$
Associated Concepts
Coming soon
FullPage
overview
significance
derivation
associated concepts