- #1

olgerm

Gold Member

- 533

- 34

In some sources QM is explained using bracket notation. I quite understand algebra of bracket notation, but I do not understand how is this notation related with physically meaningful things? How is bracket notation related to wavefunction notation?

Could you tell me whether following is true:

to express QM-system in bracket notation you:

Could you tell me whether following is true:

to express QM-system in bracket notation you:

- solve schrödinger equation

##U_{System Potential Energy}(r_1,r_2,r_3,...,r_n,t)-\sum_{n=1}^N((\frac{d^2Ψ(r_1,r_2,r_3,...,r_n,t)}{dx_n^2}+\frac{d^2Ψ(r_1,r_2,r_3,...,r_n,t)}{dy_n^2}+\frac{d^2Ψ(r_1,r_2,r_3,...,r_n,t)}{dz_n^2})*\frac{ħ^2}{m_n})=i*ħ \frac{dΨ(r_1,r_2,r_3,...,r_n,t)}{dt}##

and get wavefunction Ψ.

- apply Fourier transformation to wavefunction.##\Psi(x)=\alpha(1) \cdot sin(1 \cdot x)+\alpha(2) \cdot sin(2 \cdot x)+\alpha(3) \cdot sin(3 \cdot x)...=\sum_{k=0}^\infty(\alpha(k) \cdot sin(k \cdot x))##

or

##\Psi(x)=\alpha(1) \cdot e^{-i \cdot 1 \cdot x}+\alpha(2) \cdot e^{-i \cdot 2 \cdot x}+\alpha(3) \cdot e^{-i \cdot 3 \cdot x}...=\sum_{k=0}^\infty(\alpha(k) \cdot e^{-i \cdot k \cdot x})##

(Which one? How to make Fourier transformation to ##\Psi(t;x;y;z)##?) - make bra of Fourier transformation results. ##<\psi|=(\alpha(0);\alpha(1);\alpha(2);\alpha(3);...)##

and ket ##|\psi>=(\alpha^*(0);\alpha^*(1);\alpha^*(2);\alpha^*(3);...)##. - to find momentum of a particle solve equation ##\hat{p}|\psi>=p_x \cdot |\psi>## aka ##-i\frac{\partial}{\partial x}|\psi>=p_x \cdot |\psi>## for ##p_x##.

(in which cases the operator must be hermitian?)

Last edited: