$ \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\abs}[2][]{\left\lvert#2\right\rvert_{\text{#1}}} \newcommand{\ket}[1]{\left\lvert#1 \right.\rangle} \newcommand{\bra}[1]{\langle\left. #1\right\rvert} \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\dd}{\text{d}} \newcommand{\dv}[2]{\frac{\dd #1}{\dd #2}} \newcommand{\pdv}[2]{\frac{\partial}{\partial #1}} $
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Overview
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A Hamiltonian is time independent if $H(t)=H(0)$ for all times $t$. Time independent does not mean the state nor relative amounts of energy doesn't change. It just means the total energy of the system doesn't change (ie. it is a closed system)
The states of the Hilbert space evolve according to

Notations

Coming soon

Concepts

If $A$ is another operator and $[A, H]=0$, then $A$ is also time independent.
We can solve for schrodinger's equation for the evolution of a quantum state, given a general time-independent Hamiltonian. In general at time $t$, the state $|\psi(t)\rangle$ is $$\begin{align*} |\psi(t)\rangle = \sum_{n}c_n(0)e^{-\frac{iE_n t}{\hbar}}|E_n\rangle \end{align*}$$
We look at the probability of the quantum state being in the state $|a\rangle$, where $|a\rangle = a_1|E_1\rangle + a_2|E_2\rangle$. The probability is $$\begin{align*} |\langle a|\psi(t)\rangle|^2 &= |(a_1^*\rangle E_1|+a_2^*\rangle E_2|)(c_1 e^{-iE_1t/\hbar}|E_1\rangle + c_2e^{-iE_2t/\hbar}|E_2\rangle)|^2 \end{align*}$$\\ &=|a_1^*c_1e^{-E_1t/\hbar}+a_2^* c_2 e^{-iE_2t/\hbar}|^2\\ |a_1|^2|c_1|^2+|a_2|^2|c_2|^2+2\text{Re}[a_1c_1^*a_2^*c_2]
A Hamiltonian is time independent if $H(t)=H(0)$ for all times $t$. Time independent does not mean the state nor relative amounts of energy doesn't change. It just means the total energy of the system doesn't change (ie. it is a closed system)
The states of the Hilbert space evolve according to

Notations

Coming soon

Concepts

If $A$ is another operator and $[A, H]=0$, then $A$ is also time independent.
We can solve for schrodinger's equation for the evolution of a quantum state, given a general time-independent Hamiltonian. In general at time $t$, the state $|\psi(t)\rangle$ is $$\begin{align*} |\psi(t)\rangle = \sum_{n}c_n(0)e^{-\frac{iE_n t}{\hbar}}|E_n\rangle \end{align*}$$
We look at the probability of the quantum state being in the state $|a\rangle$, where $|a\rangle = a_1|E_1\rangle + a_2|E_2\rangle$. The probability is $$\begin{align*} |\langle a|\psi(t)\rangle|^2 &= |(a_1^*\rangle E_1|+a_2^*\rangle E_2|)(c_1 e^{-iE_1t/\hbar}|E_1\rangle + c_2e^{-iE_2t/\hbar}|E_2\rangle)|^2 \end{align*}$$\\ &=|a_1^*c_1e^{-E_1t/\hbar}+a_2^* c_2 e^{-iE_2t/\hbar}|^2\\ |a_1|^2|c_1|^2+|a_2|^2|c_2|^2+2\text{Re}[a_1c_1^*a_2^*c_2]
FullPage
Overview
Notations
Concepts
FullPage
Overview
Notations
Concepts
A Hamiltonian is time independent if $H(t)=H(0)$ for all times $t$. Time independent does not mean the state nor relative amounts of energy doesn't change. It just means the total energy of the system doesn't change (ie. it is a closed system)
The states of the Hilbert space evolve according to

Notations

Coming soon

Concepts

If $A$ is another operator and $[A, H]=0$, then $A$ is also time independent.
We can solve for schrodinger's equation for the evolution of a quantum state, given a general time-independent Hamiltonian. In general at time $t$, the state $|\psi(t)\rangle$ is $$\begin{align*} |\psi(t)\rangle = \sum_{n}c_n(0)e^{-\frac{iE_n t}{\hbar}}|E_n\rangle \end{align*}$$
We look at the probability of the quantum state being in the state $|a\rangle$, where $|a\rangle = a_1|E_1\rangle + a_2|E_2\rangle$. The probability is $$\begin{align*} |\langle a|\psi(t)\rangle|^2 &= |(a_1^*\rangle E_1|+a_2^*\rangle E_2|)(c_1 e^{-iE_1t/\hbar}|E_1\rangle + c_2e^{-iE_2t/\hbar}|E_2\rangle)|^2 \end{align*}$$\\ &=|a_1^*c_1e^{-E_1t/\hbar}+a_2^* c_2 e^{-iE_2t/\hbar}|^2\\ |a_1|^2|c_1|^2+|a_2|^2|c_2|^2+2\text{Re}[a_1c_1^*a_2^*c_2]
A Hamiltonian is time independent if $H(t)=H(0)$ for all times $t$. Time independent does not mean the state nor relative amounts of energy doesn't change. It just means the total energy of the system doesn't change (ie. it is a closed system)
The states of the Hilbert space evolve according to

Notations

Coming soon

Concepts

If $A$ is another operator and $[A, H]=0$, then $A$ is also time independent.
We can solve for schrodinger's equation for the evolution of a quantum state, given a general time-independent Hamiltonian. In general at time $t$, the state $|\psi(t)\rangle$ is $$\begin{align*} |\psi(t)\rangle = \sum_{n}c_n(0)e^{-\frac{iE_n t}{\hbar}}|E_n\rangle \end{align*}$$
We look at the probability of the quantum state being in the state $|a\rangle$, where $|a\rangle = a_1|E_1\rangle + a_2|E_2\rangle$. The probability is $$\begin{align*} |\langle a|\psi(t)\rangle|^2 &= |(a_1^*\rangle E_1|+a_2^*\rangle E_2|)(c_1 e^{-iE_1t/\hbar}|E_1\rangle + c_2e^{-iE_2t/\hbar}|E_2\rangle)|^2 \end{align*}$$\\ &=|a_1^*c_1e^{-E_1t/\hbar}+a_2^* c_2 e^{-iE_2t/\hbar}|^2\\ |a_1|^2|c_1|^2+|a_2|^2|c_2|^2+2\text{Re}[a_1c_1^*a_2^*c_2]
FullPage
Overview
Notations
Concepts