Question

1. Show that Bohr' quantization condition for angular momentum 1 = mur = n? is the same as 2* TT *s= n* i 2. Use uncertain principle ApAr = h /( 2*TT) To show that Minimum radius of Bohr's H-atom with n = 1 4TTE K- rn = n² me2 Hint: express Total energy as a function of r, find the condition for E is minimum. 3. Show that E = (-1/2)* k*e^2/r for H-atom Derive the energy expression as a function of principle quantum number n. E(n) = -((k*e^2/(2r1))(1/n^2) where rn with n = 1(above)

Answer 1

1.

$mvr=n\hbar =\frac {nh}{2\pi}$

$2\pi r =n\frac{ h}{mv}$

$\lambda =\frac{ h}{mv}$ (De Broglie relation)

$2\pi r =n\lambda$

2.

$mvr=pr=n\hbar$

$p =\frac {n\hbar}{r}$

$E =\frac {n^2\hbar^2}{2mr^2}-\frac {e^2}{4\pi \varepsilon_0 r}$

$\frac {dE}{dr}= -\frac {n^2\hbar^2 }{mr^3}+\frac {e^2}{4\pi \varepsilon _0 r^2}=0$

$\frac {n^2\hbar^2 }{mr^3}=\frac {e^2}{4\pi \varepsilon _0 r^2}$

$r_n=n^2 \frac {4\pi \varepsilon_0 \hbar^2}{me^2}$

3.

$\frac {mv^2}{r}=\frac {ke^2}{r^2}$

$mv^2=\frac {ke^2}{r}$

$E=\frac {1}{2}mv^2 -\frac {ke^2}{r}$

$E=\frac {1}{2}\frac {ke^2}{r} -\frac {ke^2}{r}$

$E=\left (-\frac {1}{2} \right )\frac {ke^2}{r}$

$E=-\frac {1}{2} \frac {ke^2}{n^2 \frac {4\pi \varepsilon_0 \hbar^2}{me^2}}$

$E=- \frac {m}{ 32\pi \varepsilon_0 \hbar^2n^2}$