The electron moves in an infinitely deep potential well with a width $ℓ = 0.35 nm$.

**a)** Calculate the electron's minimum speed $v_1$ (i.e. the ground state).

**b)** Calculate the reaction force, which is causes by electron moving back and forth and colliding to another wall of well adiabatically (thermal insulation).

**c)** Calculate the frequency of electron's reciprocating motion.

**What I did:**

**a)** $E_1=\frac{h^2}{8m_el^2} \hspace{10mm} v_1=\sqrt{\frac{2E_1}{m_e}}$

So I got this answer, which is correct:

$E_1=4.9176...\times10^{-19}\ J$

$v_1=1.0390...\times10^6\ m/s\approx1.0\times10^6\ m/s$

**b)** I tried this formula:

$F_r=\frac{m_ev^2}{r}\hspace{10mm}r_1=\frac{h^2\epsilon_0}{\pi m_ee^2}$

But I didn't get correct answer.

**c)** $f=\frac{2E_1}{h}$

So I got this answer, which is correct:

$f=1.4843\times 10^{15}\frac{1}{s}\approx 1.5\times 10^{15}\frac{1}{s}$

**So, my question is what formula should I use to get the correct force ($F$ ) value?

Because obviously the formula which I chose is incorrect.

**EDIT:**

I used this formula and got correct answer:

$F=\frac{\triangle p}{\triangle t}=\frac{2m_ev_1}{2\frac{l}{v_1}}=m_ev_1\times\frac{v_1}{l}=\frac{m_ev_1^2}{l}\approx 2.8\times 10^{-9}\ N$

So is it correct approach? Or I was just lucky and got correct answer?