a, Ta có:

Hai hàm sóng trực giao nhau khi  \(I=\int\psi_{1s}.\psi_{2s}d\psi=0\) \(\Leftrightarrow I=\iiint\psi_{1s}.\psi_{2s}dxdydz=0\)

Chuyển sang tọa độ cầu ta có:  \(\begin{cases}x=r.\cos\varphi.sin\theta\\y=r.\sin\varphi.sin\theta\\z=r.\cos\theta\end{cases}\)

\(\Rightarrow\)\(I=\frac{a^3_o}{4.\sqrt{2.\pi}}\int\limits^{\infty}_0\left(2-\frac{r}{a_o}\right).e^{-\frac{3.r}{2.a_o}}.r^2.\sin\theta dr\int\limits^{2\pi}_0d\varphi\int\limits^{\pi}_0d\theta\)

       \(=a^3_o.\sqrt{\frac{\pi}{2}}\)(.\(2.\int\limits^{\infty}_0r^2.e^{-\frac{3.r}{2.a_o}}dr-\frac{1}{a_o}.\int\limits^{\infty}_0r^3.e^{-\frac{3.r}{2.a_o}}dr\))

         \(=a_o.\sqrt{\frac{\pi}{2}}.\left(2.I_1-\frac{1}{a_o}.I_2\right)\)  

Tính \(I_1\):

Đặt \(r^2=u\); \(e^{-\frac{3r}{2a_o}}dr=dV\)

\(\Rightarrow\begin{cases}2.r.dr=du\\-\frac{2a_o}{3}.e^{-\frac{3r}{2a_o}}=V\end{cases}\)    \(\Rightarrow I_1=-r^2.\frac{2a_o}{3}.e^{-\frac{3r}{2a_o}}+\frac{4.a_o}{3}.\int\limits^{\infty}_0r.e^{-\frac{3r}{2a_o}}dr\)\(=0+\frac{4a_o}{3}.I_{11}\)

Tính \(I_{11}\):

Đặt r=u; \(e^{-\frac{3r}{2a_o}}dr=dV\)\(\Rightarrow\begin{cases}dr=du\\-\frac{2a_o}{3}.e^{-\frac{3r}{2a_o}}=V\end{cases}\)\(\Rightarrow I_{11}=0+\frac{2a_0}{3}.\int\limits^{\infty}_0e^{-\frac{3r}{2a_o}}dr=\frac{4a^2_o}{9}\)

\(\Rightarrow2.I_1=2.\frac{4a_o}{3}.\frac{4a_o^2}{9}=\frac{32a^3_o}{27}\)

Tính \(I_2\):

Đặt \(r^2=u;e^{-\frac{3r}{2a_o}}dr=dV\) \(\Rightarrow\)\(3r^2dr=du;-\frac{2a_o}{3}.e^{-\frac{3r}{2a_o}}=V\)

\(\Rightarrow I_2=0+2.a_o.\int\limits^{\infty}_0r^2.e^{-\frac{3r}{2a_o}}dr\)\(\Rightarrow\frac{1}{a_o}.I_2=2a_o.\frac{16a^3_o}{27}.\frac{1}{a_o}=\frac{32a^3_o}{27}\)

\(\Rightarrow I=a^3_o.\sqrt{\frac{\pi}{2}}.\left(\frac{32a^3_o}{27}-\frac{32a^3_o}{27}\right)=0\)

Vậy hai hàm sóng này trực giao với nhau.

b,

Xét hàm \(\Psi_{1s}\):

Hàm mật độ sác xuất là: \(D\left(r\right)=\Psi^2_{1s}=\frac{1}{\pi}.a^3_o.e^{-\frac{2r}{a_o}}\)

\(\Rightarrow D'\left(r\right)=-\frac{2.a_o^2}{\pi}.e^{-\frac{2r}{a_o}}=0\)

\(\Rightarrow\)Hàm đạt cực đại khi \(r\rightarrow o\) nên hàm sóng có dạng hình cầu.

Xét hàm \(\Psi_{2s}\):

Hàm mật độ sác xuất: \(D\left(r\right)=\Psi_{2s}^2=\frac{a^3_o}{32}.\left(2-\frac{r}{a_o}\right)^2.e^{-\frac{r}{a_0}}\)\(\Rightarrow D'\left(r\right)=\left(2-\frac{r}{a_o}\right).e^{-\frac{r}{a_o}}.\left(-4+\frac{r}{a_o}\right)=0\)

\(\Rightarrow r=2a_o\Rightarrow D\left(r\right)=0\); \(r=4a_o\Rightarrow D\left(r\right)=\frac{a^3_o}{8}.e^{-4}\)

Vậy hàm đạt cực đại khi \(r=4a_o\), tại \(D\left(r\right)=\frac{a^3_o}{8}.e^{-4}\)

hai hàm trực giao: I=\(\int\)\(\Psi\)*\(\Psi\)d\(\tau\)=0

Ta có: I=\(\int\limits^{ }_x\)\(\int\limits^{ }_y\)\(\int\limits^{ }_z\)\(\Psi\)*\(\Psi\)dxdydz=0

           =\(\int\limits^{ }_r\)\(\int\limits^{ }_{\theta}\)\(\int\limits^{ }_{\varphi}\)\(\Psi\)_1s\(\Psi\)_2sr²sin\(\theta\)drd\(\theta\)d\(\varphi\)

          =\(\int\limits^{\infty}_0\)\(\int\limits^{\pi}_0\)\(\int\limits^{2\pi}_0\)(2-\(\frac{r}{a_0}\)).e^-3r/a₀r²sin\(\theta\)drd\(\theta\)d\(\varphi\)

          =C.\(\int\limits^{\infty}_0\)(2-\(\frac{r}{a_0}\)).e^-3r/a₀r2dr.\(\int\limits^{\pi}_0\)sin\(\theta\)\(\int\limits^{2\pi}_0\)d\(\varphi\)

 với C=\(\frac{1}{4\sqrt{2\pi}}\)a₀^-3

 Xét tích phân: J=\(\int\limits^{\infty}_0\)(2-\(\frac{r}{a_0}\)).e^-3r/a₀r²dr

 =\(\int\limits^{\infty}_0\)(2r²- \(\frac{r^3}{a_0}\)).e^-3r/a₀dr

 =\(\int\limits^{\infty}_0\)(2r²- \(\frac{r^3}{a_0}\)).\(\frac{-2a_0}{3}\)de^-3r/a₀

  =\(\frac{-2a_0}{3}\).((2r²-\(\frac{r^3}{a_0}\))e^-3r/a₀\(-\)\(\int\)(4r-\(\frac{3r^2}{a_0}\))e^-3r/adr)

 =\(\frac{-2a_0}{3}\)((2r²-\(\frac{r^3}{a_0}\))e^-3r/a₀ - \(\int\)(4r-\(\frac{3r^2}{a_0}\)).\(\frac{-2a_0}{3}\)de^-3r/a)

 =\(\frac{-2a_0}{3}\)((2r²-\(\frac{r^3}{a_0}\))e^-3r/a₀ +\(\frac{2a_0}{3}\).((4r-\(\frac{3r^2}{a_0}\))e^-3r/a₀ - \(\int\)(4 - \(\frac{6r}{a_0}\))e^-3r/a₀dr))

 =\(\frac{-2a_0}{3}\)((2r²-\(\frac{r^3}{a_0}\))e^-3r/a₀ +\(\frac{2a_0}{3}\).((4r-\(\frac{3r^2}{a_0}\))e^-3r/a₀- \(\int\)(4 - \(\frac{6r}{a_0}\))\(\frac{-2a_0}{3}\).de^-3r/a₀))

 =\(\frac{-2a_0}{3}\)(((2r²-\(\frac{r^3}{a_0}\))e^-3r/a₀ +\(\frac{2a_0}{3}\).((4r-\(\frac{3r^2}{a_0}\))e^-3r/a₀+\(\frac{2a_0}{3}\)((4-\(\frac{6r}{a_0}\)).e^-3r/a₀ + \(\int\)(\(\frac{6}{a_0}\)e^-3r/a₀dr)))

=\(\frac{-2a_0}{3}\)(((2r²-\(\frac{r^3}{a_0}\))e^-3r/a₀ +\(\frac{2a_0}{3}\).((4r-\(\frac{3r^2}{a_0}\))e^-3r/a₀+\(\frac{2a_0}{3}\)((4-\(\frac{6r}{a_0}\)).e^-3r/a₀ + \(\int\)(\(\frac{6}{a_0}\).\(\frac{-2a_0}{3}\)de^-3r/a₀)))

=\(\frac{-2a_0}{3}\)((((2r²-\(\frac{r^3}{a_0}\))e^-3r/a₀ +\(\frac{2a_0}{3}\).((4r-\(\frac{3r^2}{a_0}\))e^-3r/a₀+\(\frac{2a_0}{3}\)((4-\(\frac{6r}{a_0}\)).e^-3r/a₀ - 4.e^-3r/a₀))))

b1.\(\Psi\)_1s=\(\frac{1}{\sqrt{\pi}}\).a₀^3/2.e^-r/a₀.

Hàm mật độ xác suất :

      D_r=|\(\Psi\)²|r²

              =\(\frac{1}{\pi}\)a₀³.e^-2r/a₀.r²

xét \(\frac{dD_r}{r}\)= \(\frac{1}{\pi}\)a₀³.(r².\(\frac{-2}{a_0}\)e^-2r/a₀+2r.e^-2r/a₀)

           = \(\frac{1}{\pi}\)a₀^3.e^-2r/a₀.2r.(1\(-\)\(\frac{r}{a_0}\))

        \(\frac{dD_r}{r}\)=0 \(\Leftrightarrow\)r=a₀.

tại r=a₀ D_r đạt cực đại. D_max=\(\frac{1}{\pi}\)a₀³.e^-2.a₀²=\(\frac{1}{\pi}\)a₀⁵.e^-2

b2. \(\Psi\)_2s=\(\frac{1}{4\sqrt{2}}\)a₀^3/2.(2-\(\frac{r}{a_0}\)).e^-r/2a_o.

Hám mật độ xác suất:

    D_r=|\(\Psi\)²|r².

       =\(\frac{1}{32}\).a₀³.e^-r/a_o.(2-\(\frac{r}{a_0}\))²

       =\(\frac{1}{32}\).a₀³.e^-r/a_o.(4-4.\(\frac{r}{a_0}\)+\(\frac{r^2}{a^2_0}\))

  Xét \(\frac{dDr}{dr}\)= \(\frac{1}{32}\).a₀³.((e^-r/a_o.\(\frac{-1}{a_0}\).(2-\(\frac{r}{a_0}\))²+e^-r/a_o.(-.\(\frac{4}{a_0}\)+\(\frac{2r}{a^2_0}\))

              =- \(\frac{1}{32}\).a₀³.e^-r/a_o.\(\frac{1}{a_0}\).(2-\(\frac{r}{a_0}\))(\(\frac{r}{a_0}\)-4)

a) Ta có:   Mật độ xác suất tìm thấy electron trong vùng không gian xung quanh hạt nhân nguyên tử:

    D(r) = R²(r) . r²

             = 416/729 . a₀^-5 . r² . (2 - r/3a₀)² . e^-2r/3a₀ . r²

           = 416/729 . a₀^-5 . (4r⁴ - 4r⁵/3a₀ + r⁶/9a₀²) .  e^-2r/3a₀

      Khảo sát hàm số D(r) thuộc r

          Xét:  d D(r)/ dr = 416/729 . a₀^-5 . [(16r³ - 20r⁴/3a₀ + 2r⁵/3a_0²) .  e^-2r/3a₀  -  (4r⁴ - 4r⁵/3a₀ + r⁶/9a_0²) . 2/3a₀  . e^-2r/3a₀ ]

                          = 416/729 . a₀^-5 . e^-2r/3a₀  . r³ . (16a₀³ - 28r/3a₀ + 14r²/9a₀² - 2r³/27a₀³)

                          = 832/19683 . a₀^-8 . e^-2r/3a₀  . r³ . (-r³ +21r².a₀ - 126r.a₀² +216a₀³)

                          = - 832/19683 . a₀^-8 . e^-2r/3a₀  . r³ . (r - 6a₀).(r - 3a₀).(r - 12a₀)

           d D(r)/ dr = 0. Suy ra r =0; r =3a₀ ; r = 6a₀; r = 12a₀

           Với r = 0 : D(r) =0

                  r =3a₀ : D(r) = 416/9 .a^-1 . e^-2

                  r =6a₀ : D(r) = 0

                  r =12a₀ : D(r) = 425984/9.a^-1 . e^-8

^{b) Ai vẽ câu này rồi cho   up lên với, cám ơn mọi người trước nhé!}  

a)Mật độ xác suất có mặt electron tỷ lệ với |R_3P|².r²

D_(r)=|R_3P|².r²  =D _(r)=\(\frac{416}{729}\) .a₀^-5.(2r²- \(\frac{r^3}{3a_0}\)).\(^{e^{-\frac{2r}{3a_0}}}\)

   Lấy đạo hàm của D theo r để khảo sát mật độ xác suất :

    D' _(r)= \(\frac{416}{729}\) .a₀^-5.2.(2r²-\(\frac{r^3}{3a_0}\)).(4r-\(\frac{r^2}{a_0}\)).\(^{e^{-\frac{2r}{3a_0}}}\)+\(\frac{416}{729}\) .a₀^-5.(2r²-\(\frac{r^3}{3a_0}\))².(-\(\frac{2}{3a_0}\)).\(^{e^{-\frac{2r}{3a_0}}}\) 

           =\(\frac{832}{729}\). a₀^-6.\(^{e^{-\frac{2r}{3a_0}}}\). (2r²-\(\frac{r^3}{3a_0}\)) .[(4r-\(\frac{r^2}{a_0}\)).a₀ -\(\frac{1}{3}\). (2r²-\(\frac{r^3}{3a_0}\))]

            =\(\frac{832}{729}\). a₀^-6.\(^{e^{-\frac{2r}{3a_0}}}\).r³.(2- \(\frac{r}{3a_0}\)).(\(\frac{r^2}{9a_0}-\frac{5r}{3}+4a_0\))

=>D’_(r)=0   => r=0 ,r=3a₀ ,r=6a₀ ,r=12a₀.

Với:r=0      =>D_(r)=0

       r=3a₀  =>D_(r)=0

       r=6a₀  =>D_(r)=\(\frac{416}{9a_0.e^2}\)

       r=12a₀=>D_(r)=\(\frac{425984}{a_0.e^8}\)

b)

a)Mật  độ xác suất :D_(r)=|R_3P|².r²  =>D _(r)= .a₀^-5.(2r²-)².                                                                                                                                             Ta có:D^’_(r)= .a₀^-5.2.(2r²-).(4r-).+ .a₀^-5.(2r²-)²..(-).

=>D’_(r)=. a₀^-6.. (2r²-) .[(4r-).a₀ -. (2r²-)]

D’_(r)=0    =>(2r²-) .[(4r-).a₀ -. (2r²-)]=0  => r=0 ,r=3a₀ ,r=6a₀ ,r=12a₀.

Với:r=0     =>D_(r)=0

       r=3a₀  =>D_(r)=0

       r=6a₀  =>D_(r)=

       r=12a₀=>D_(r)=

Đáp án D

Ta có Đáp án D

Ta có y’ = –f’(1 – x) + 2018 = –[1–(1–x)][(1–x)+2]g(1–x) – 2018 + 2018

= –x(3–x)g(1–x)

Suy ra  (vì g(1–x) < 0,  ∀ x ∈ R ) 

Vậy hàm số đã cho nghịch biến trên khoảng  3 ; + ∞

Đáp án là D