양자물리 3. 수식화

이번 포스트는 영어로 작성하겠다.
수식 기호들과 한글을 번갈아 가면서 쓰니까 손이 너무 바빠져서..
일단 이번 포스트만 영어로 써보고 편하면 앞으로도 영어로 쓸까 한다.
영어가 다소 미숙하나 이것도 writing 연습도 할겸..

In this chapeter, I will introduce some methematic notations and concepts

Hilbert space

Hilbert space is set of functions that follow

Physicist say it Hilbert space and mathematician say it $L^2(a,b)$

In Hibert space, if any function f(x) can be expressed as $f(x)=\Sigma _{n=1} ^\infty f_n(x)$, then function set of $f_n(x)$ is called complete set

in the situation above, $f$ and $f_n$ follow $c_n=<f|f_n>$, which is according to fourier method

Hermitian operator

All observations is real value, so that kind of operation must be hermitian. and we say it Hermitian operator which follows

or

observatoin result is expressed as real value

Eigen fn of Hermitian operator

if only one value comes out for every observation Q, this state is called decided state.
And for Hamiltonian operator, this state is called normal state
This situation can be expressed as

At normal state, every observation for particle of $\Psi$ gives permitted energy value $E_n$
That is eigenvalue of Hermitian operator and the set of eigenvalue is clalled spectrum.

Discontinuous eigen fn

For discontinuous case,

rule1. eigenvalue is real value
rule2. eigen fn of different eigen value is orthogonal
rule3. axion) for every hermitian operator, eigen fn is complete set.

Continuous eigen fn

For continuous, eigen fn can't normalized and do not follow rule 1,2.
I will show this with example of momentum operator

momentum operator $\hat p$is expressed as $-i\hbar\frac{d}{dx}$ and if we say eigen fn as $f_p(x)$,

Here p is eigenvalue and momentum value.
The normalized general sol is

and $\langle f_{p'}|f_p\rangle = \delta(p-p')$
so $f_p$ is orthogonal
and as it is complete set,

here c(p) is amplitude of that momentum value (p) which can be called $\varphi(p)$
and $c_n$can be calculate by

To sum up,

Eigen fn can not normalized and is not always real eigen value
But if it habe real eigen value, eigen fn can be orthonormalized and make complete set

Fourier TR and Dirac notation

At above, the state function at position-space and momentum-space are
followings.

$\psi(x)=\int _{-\infty} ^\infty c(p)f_p (x) dp$

$=\frac{1}{\sqrt{2\pi \hbar}}\int_{-\infty}^{\infty}\varphi(p)e^{ipx/\hbar}dp$

and

$\varphi(p)= \langle f_{p}|f\rangle$

$=\frac{1}{\sqrt{2\pi \hbar}}\int_{-\infty}^{\infty}\psi(p)e^{-ipx/\hbar}dx$

which is state description at momentum space.