my Quantum Mechanics cheat sheet

Table 1: QM cheat sheet


	Position Operator \(X\)	Momentum operator \(P\)	Hamilitonian 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	1. Probability of measuring \(x\) given system is in state \(\ket {\Psi }\) is \(\|\braket {\Psi \|\Psi }\|^2\). For inﬁnite potential deep well of width \(x<0<L\) this becomes \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*} 2. Probability of measuring \(x\) given system is in state \(\phi _p\)	\(\ket {\Psi } = \int _{-\infty }^{\infty } \ket {\phi _p} \braket {\phi _p\|\Psi } \,dp\)	\(\ket {\Psi } = \int _{-\infty }^{\infty } \ket {E_i} \braket {E_i\|\Psi } \,di\)