💇🏻 💺 🎅🏽 Putting all the dots on the "psi" 💚 👠 🥘

When planning several articles one way or another related to quantum mechanics, it was decided to put a discussion of a number of technical issues, philosophical disputes and idle myths into a separate article. It will be about the most complex and interesting tool of human intelligence - quantum theory.

I'm not a physicist at all, but I know what's what.
Popeye the sailor

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( ) - : $\phi' = T\phi$ . . : "" .

U, $\Phi = USU^*$ — - S , , . , $\mid\psi\rangle$ $U\mid\psi\rangle$ .

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$U_tU_s = U_{t+s}$ ( )
$t\rightarrow U_t$

U_{t} = e^{- i t H}

$U_t = e^{-itH}$

H — , t . , $\mid\psi_t\rangle = U_t \mid\psi\rangle$

i ℏ \frac{\partial | Ψ ⟩}{\partial t} = \hat{H} | Ψ ⟩

$i\hbar\frac{\partial\left|\Psi\right>}{\partial t} = \hat{H}\left|\Psi\right>$

: $\hat H = \hat T + \hat U$ — , , , $\hat T = \sum^N_k\frac{\hat p^2_k}{2m_k}$ .

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\hat{H} ψ_{x} = E ψ_{x}

$\hat{H}\psi_x = E\psi_x$

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ψ (x) = C^{+} \cos k x + C^{-} \sin k x

$\psi(x) = C^+\cos kx+ C^-\sin kx$

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ψ_{k} (x, E) = A_{k} e^{i γ_{k} x} + B_{k} e^{- i γ_{k} x} γ_{k} (E) = \sqrt{\frac{2 m_{k} (E - U_{k})}{ℏ^{2}}}

$\psi_k(x,E) = A_k e^{i\gamma_k x} + B_k e^{-i\gamma_k x} \\ \gamma_k(E) = \sqrt{\frac{2m_k(E-U_k)}{\hbar^2}}$

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i \frac{ψ (\vec{r}, t + Δ t) - ψ (\vec{r}, t)}{Δ t} = \frac{\hat{H} ψ (\vec{r}, t) + \hat{H} ψ (\vec{r}, t + Δ t)}{2}

$i \frac{\psi(\vec{r},t+\Delta t) - \psi(\vec{r},t)}{\Delta t} = \frac{\hat{H}\psi(\vec{r},t) + \hat{H}\psi(\vec{r},t+\Delta t)} {2}$

[1 + \frac{i Δ t}{2} \hat{H}] ψ (\vec{r}, t + Δ t) = [1 - \frac{i Δ t}{2} \hat{H}] ψ (\vec{r}, t)

$\left[1 + \frac{i\Delta t}{2}\hat{H}\right]\psi(\vec{r},t+\Delta t) = \left[1 - \frac{i\Delta t}{2}\hat{H}\right]\psi(\vec{r},t)$

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\hat{H} ψ (x, t) = - \frac{ψ (x + Δ x, t) + ψ (x - Δ x, t) - 2 ψ (x, t)}{Δ x^{2}} + V (x) ψ (x, t)

$\hat{H}\psi(x,t) = - \frac{\psi(x+\Delta x,t) + \psi(x-\Delta x,t) - 2\psi(x,t)} {\Delta x^2} + V(x)\psi(x,t)$

. , $N_x = 5$ :

H^{1 D} = (\begin{matrix} - 2 & 1 & 0 & 0 & 0 \\ 1 & - 2 & 1 & 0 & 0 \\ 0 & 1 & - 2 & 1 & 0 \\ 0 & 0 & 1 & - 2 & 1 \\ 0 & 0 & 0 & 1 & - 2 \end{matrix}), ψ^{1 D} = (\begin{matrix} ψ_{1} \\ ψ_{2} \\ ψ_{3} \\ ψ_{4} \\ ψ_{5} \end{matrix})

$H^{1D}=\begin{pmatrix} -2 & 1 & 0 & 0 & 0\\ 1 & -2 & 1 & 0 & 0\\ 0 & 1 & -2 & 1 & 0\\ 0 & 0 & 1 & -2 & 1\\ 0 & 0 & 0 & 1 & -2 \end{pmatrix}, \qquad\qquad \psi^{1D}=\begin{pmatrix} \psi_1\\ \psi_2\\ \psi_3\\ \psi_4\\ \psi_5 \end{pmatrix}$

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H (H_{2} O) = \sum_{α = 1}^{3} \frac{p_{α}^{2}}{2 M_{α}} + \sum_{i = 1}^{10} \frac{p_{i}^{2}}{2 m_{e}} - \sum_{α = 1}^{3} \sum_{i = 1}^{10} \frac{Z_{α} e^{2}}{| R_{α} - r_{i} | 4 π ε_{0}} + + \sum_{α, β, α < β}^{3} \frac{Z_{α} Z_{β}}{| R_{α} - R_{β} | 4 π ε_{0}} + \sum_{i, j, j < j}^{10} \frac{e^{2}}{r_{i} - r_{j} | 4 π ε_{0}}

$H\left(H_{2} O\right)=\sum_{\alpha=1}^{3} \frac{p_{\alpha}^{2}}{2 M_{\alpha}}+\sum_{i=1}^{10} \frac{p_{i}^{2}}{2 m_{e}}-\sum_{\alpha=1}^{3} \sum_{i=1}^{10} \frac{Z_{\alpha} e^{2}}{\left|R_{\alpha}-r_{i}\right| 4 \pi \varepsilon_{0}}+\\ +\sum_{\alpha, \beta, \alpha<\beta}^{3} \frac{Z_{\alpha} Z_{\beta}}{\left|R_{\alpha}-R_{\beta}\right| 4 \pi \varepsilon_{0}}+\sum_{i, j, j<j}^{10} \frac{e^{2}}{r_{i}-r_{j} | 4 \pi \varepsilon_{0}}$

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It will be difficult, but worth it. No matter what they say about religious texts (by the way, it was the Bible that killed a Christian in me), about how they put everything in its place and give answers to hidden questions, they cannot be compared with the beauty of mathematics of various formalisms, with the intricate interconnection of the sections of physics, with explanatory the power of evolutionary theory and the simplicity of fundamental principles that give rise to a variety of universal scales. Cognition opens to the eye more and more beauty of the world. Isn’t that the meaning of life?

Putting all the dots on the "psi"

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