Physics: Para compreendermos a carta de João da Silva, precisamos identificar o contexto que o levou a escrever a empresa Computerly Ltda. Que contexto é esse?

First partThe wavefunction isAs we know, the wavefunction is normalized if:Lets prove it:Lets take the first integralwe can use the substitutionwhen x= 0 then y = 0, and when , inverting the limits of integration to take care of the minus signbut this is the second integral!!!sonow, we can see thatsoThis is what we wanted to show!Second partWe just need to obtain :We can use the same approach:...

Para compreendermos a carta de João da Silva, precisamos

Ответ:

sakhya04

29.04.2023

Physics

First part

The wavefunction is

$\psi = \left \{ {{e^{-x} \ \ x\ge0} \atop {e^{x} \ \ x$

As we know, the wavefunction is normalized if:

$\int\limits^{\infty } _{-\infty } {|\psi(x)|^2} \, dx =1$

Lets prove it:

$\int\limits^{\infty } _{-\infty } {|\psi(x)|^2} \, dx = \int\limits^{0 } _{-\infty } {|\psi(x)|^2} \, dx + \int\limits^{\infty } _{0 } {|\psi(x)|^2} \, dx$

$\ \int\limits^{0 } _{-\infty } {(e^{x} )^2} \, dx + \int\limits^{\infty } _{0 } {(e^{-x}) ^2} \, dx$

Lets take the first integral

$\ \int\limits^{0 } _{-\infty } {(e^{x} )^2} \, dx$

we can use the substitution

$x = -y \\\\dx = - dy$

when x= 0 then y = 0, and when $x= - \infty$ , $y=\infty$

$\ \int\limits^{0 } _{\infty } {(e^{-y} )^2} \, (-dy)$

$\ - \int\limits^{0 } _{\infty } {(e^{-y} )^2} \, dy$

inverting the limits of integration to take care of the minus sign

$\ \int\limits^{ \infty} _{0 } {(e^{-y} )^2} \, (dy)$

but this is the second integral!!!

$\int\limits^{\infty } _{-\infty } {|\psi(x)|^2} \, dx = 2 * \int\limits^{\infty } _{0 } {{e^{-2x} } \, dx$

now, we can see that

$\frac{d}{dt} e^{-2x} = -2 e^{-2x}$

$\frac{1}{-2} \frac{d}{dt} e^{-2x} = e^{-2x}$

$\int\limits^{\infty } _{-\infty } {|\psi(x)|^2} \, dx = 2 * \int\limits^{\infty } _{0 } { \frac{1}{-2} * \frac{d}{dt} e^{-2x} } \, dx$

$\int\limits^{\infty } _{-\infty } {|\psi(x)|^2} \, dx = \frac{2}{-2} * \int\limits^{\infty } _{0 } { \frac{d}{dt} e^{-2x} } \, dx$

$\int\limits^{\infty } _{-\infty } {|\psi(x)|^2} \, dx = -1 * [ \right e^{-2x} \left ] ^{\infty } _{0 }$

$\int\limits^{\infty } _{-\infty } {|\psi(x)|^2} \, dx = -1 * [ ( \lim_{x\rightarrow \infty } e^{-2x} ) -e^{-2*0} ]$

$\int\limits^{\infty } _{-\infty } {|\psi(x)|^2} \, dx = -1 * ( 0 - 1)$

$\int\limits^{\infty } _{-\infty } {|\psi(x)|^2} \, dx = 1$

This is what we wanted to show!

Second part

We just need to obtain $p(|x| \le 0.5)$ :

$p(|x| \le 0.5) = \int\limits^{0.5 } _{-0.5 } {|\psi(x)|^2} \, dx$

We can use the same approach:

$\int\limits^{0.5 } _{-0.5 } {|\psi(x)|^2} \, dx = \int\limits^{0 } _{-0.5 } {|\psi(x)|^2} \, dx + \int\limits^{0.5 } _{0 } {|\psi(x)|^2} \, dx$

$\ \int\limits^{0 } _{-0.5 } {(e^{x} )^2} \, dx + \int\limits^{0.5 } _{0 } {e^{-x} ^2} \, dx$

$p(|x| \le 0.5) = 2 * \int\limits^{0.5} _{0 } {{e^{-2x} } }\, dx$

$p(|x| \le 0.5) = 2 * \int\limits^{0.5} _{0 } {\frac{1}{-2} \frac{d}{dt} e^{-2x} } \, dx$

$p(|x| \le 0.5) = -1 * [ \right e^{-2x} \left ] ^{0.5} _{0 }$

$p(|x| \le 0.5) = -1 * (e^{-1} - 1)$

$p(|x| \le 0.5) = 0.6321$

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Para compreendermos a carta de João da Silva, precisamos identificar o contexto que o levou a escrever a empresa Computerly Ltda. Que contexto é esse?

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Para compreendermos a carta de João da Silva, precisamos identificar o contexto que o levou a escrever a empresa Computerly Ltda. Que contexto é esse?