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| Position Operator \(X\) | Momentum operator \(P\) | Hamiltonian operator \(H\) | |
| Eigenvalue
eigenvector
relation |
\(X \ket {x} = x \ket {x}\) where \(x\) is the eigenvalue (size) of the \(\ket {x}\) which is the position vector associated with \(x\) measured. | \(P \ket {\phi _p} = p \ket {\phi _p}\) where \(p\) is the momentum of the particle. | \(H \ket {\Psi _{E_i}} = E_i \ket {\Psi _{E_i}}\) where \(E_i\) is the energy level of the particle. |
| Normalization relation | \(\int _{-\infty }^{\infty } \ket {x} \bra {x} \,dx = 1\) | \(\int _{-\infty }^{\infty } \ket {\phi _p} \bra {\phi _p} \,dp = 1\) | \(\int _{-\infty }^{\infty } \ket {\Psi _{E_i}} \bra {\Psi _{E_i}} \,dE = 1\) |
| orthogonality | \(\braket {x|x'}=\delta {(x-x')}\) | \(\braket {\phi _p|\phi _{p'}}=\delta {(p-p')}\) | \(\braket {\Psi _{E_i}|\Psi _{E_j}}=\delta {(E_i-E_j)}\) |
| Matrix
element
of
operator |
\(\Braket {x|X|x'}= x' \delta {(x-x')}\). Operator \(X\) is diagonal matrix. | \(\braket {x|P|x'}=-i \hbar \delta {(x-x')} \frac {d}{d x'}\) where momentum operator \(P\) is expressed in position operator \(\ket {x}\) basis. Note that operator \(P\) is not a diagonal matrix. | \(\Braket {x|H|x'}= ?\) |
| Function
form
of
the
state
function
\(\ket {\Psi }\) |
N/A ? | \begin{align*} P \ket {\phi _p} &= p \ket {\phi _p}\\ \int P \ket {x'} \braket {x'|\phi _p} \, dx &= p \int \ket {x'} \braket {x'|\phi _p} \, dx\\ \int \braket {x|P|x'} \braket {x'|\phi _p} \, dx &= p \int \braket {x|x'} \braket {x'|\phi _p} \, dx\\ \int -i \hbar \delta {(x-x')} \frac {d}{d x'} \phi _p(x') \, dx &= p \int \delta {(x-x')} \phi _p(x') \, dx\\ &= \frac {2}{L} \frac {L}{2}\\ &=1 \end{align*} |
\begin{align*} \braket {\Psi |\Psi } &= \int _{-\infty }^{\infty } \braket {\Psi |x} \braket {x|\Psi } \,dx\\ &= \int _{-\infty }^{\infty } \braket {x|\Psi } \braket {x|\Psi } \,dx\\ &= \int _{-\infty }^{\infty } \Psi ^*(x) \Psi (x) \,dx\\ &= \int _{0}^{L} \left ( \sqrt {\frac {2}{L}} \sin {\frac {n \pi x}{L}} \right )^2 \,dx\\ &= \frac {2}{L} \frac {L}{2}\\ &=1 \end{align*} |
| Vector
form
to
function
form |
\(\Braket {x|\Psi }= \Psi (x)\) | \(\Braket {x|\phi _p}= \phi _p(x)\) | \(\Braket {x|\Psi _{E}}=\Psi _{E}(x)\) |
| Expansion
of
state
vector
\(\ket {\Psi }\) |
\(\ket {\Psi } = \int _{-\infty }^{\infty } \ket {x} \braket {x|\Psi } \,dx\) | \(\ket {\Psi } = \int _{-\infty }^{\infty } \ket {\phi _p} \braket {\phi _p|\Psi } \,dp\) | \(\ket {\Psi } = \int _{-\infty }^{\infty } \ket {E_i} \braket {E_i|\Psi } \,di\) |
| State
function
\(\ket {\Psi }\)
For
infinite
potential
deep
well
of
width
\(x<0<L\) |
todo | todo | todo |
| Probability
of
measurement |
|
\(\ket {\Psi } = \int _{-\infty }^{\infty } \ket {\phi _p} \braket {\phi _p|\Psi } \,dp\) | \(\ket {\Psi } = \int _{-\infty }^{\infty } \ket {E_i} \braket {E_i|\Psi } \,di\) |