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Homework Statement
Incomming photon gives half of its energy to an electron during scatering. After scattering, photon is headed ##\phi=120^\circ## according to the original direction. What is the ##\lambda## of the incomming photon?
Homework Equations
\begin{align}
\Delta \lambda &= \frac{h}{m_ec}(1-\cos\phi)\\
W_{before} &= W_{after}
\end{align}
The Attempt at a Solution
I first calculated the ##\Delta \lambda = \frac{h}{m_e c} (1 - \cos \phi) = 3.65pm## and then wrote the energy conservation equation which i evolved into an quadratic equation:
\begin{align}
W_{before} &=W_{after}\\
W_f + m_ec^2 &= W_f' +W_e\\
W_f + m_ec^2 &= W_f' +\tfrac{1}{2}W_f\longleftarrow\substack{\text{I used the fact that photon gives}\\\text{half of its energy to an electron} }\\
m_e c^2 &= W_f' - \tfrac{1}{2}W_f\\
m_ec^2 &= hc \left(\frac{1}{\lambda'} - \frac{1}{2\lambda}\right)\\
m_ec^2 &= hc \left(\frac{1}{\lambda + \Delta \lambda} - \frac{1}{2\lambda}\right)\longleftarrow\substack{\text{since i know $\Delta \lambda$ which }\\\text{is defined as $\Delta \lambda = \lambda' - \lambda$}}\\
\frac{m_ec}{h} &= \frac{1}{\lambda + \Delta \lambda} - \frac{1}{2\lambda}\\
\frac{m_ec}{h} &= \frac{2\lambda - (\lambda + \Delta \lambda)}{2\lambda (\lambda + \Delta \lambda)}\\
\frac{2m_ec}{h} &= \frac{\lambda - \Delta \lambda}{\lambda^2 + \Delta \lambda \lambda}\\
\frac{2m_ec}{h} \lambda^2 + \frac{2m_ec}{h} \Delta \lambda \lambda &= \lambda - \Delta \lambda\\
\frac{2m_ec}{h} \lambda^2 + \left(\frac{2m_ec}{h} \Delta \lambda - 1 \right)\lambda + \Delta \lambda &=0\\
\underbrace{\lambda^2}_{A\equiv 1} + \underbrace{\left(\Delta \lambda - \frac{h}{2m_ec} \right)}_{\equiv B}\lambda + \underbrace{\frac{h}{2m_ec}\Delta \lambda}_{\equiv C} &=0\longleftarrow\substack{\text{I finaly got the quadratic equation}\\\text{for whom i define $A$, $B$ and $C$}}\\
\end{align}
Now i use the basic quadratic equation formula derived from "compleeting the square", and as you can see the number under the square root is negative. Does this mean there is no solution to this problem? If this is not the case did i miss something?
\begin{aligned}
\lambda &= \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}\\
\lambda &= \frac{ \left( \frac{h}{2m_ec} - \Delta \lambda \right) \pm \sqrt{ \left(\Delta \lambda - \frac{h}{2m_ec} \right)^2 - 4\frac{h}{2m_ec}\Delta \lambda} }{ 2 }\\
\lambda &= \frac{-2.43\times10^{-12}m \pm \sqrt{5.92\times10^{-24}m^2 - 1.78\times10^{-23}m^2}}{2}\\
\lambda &= \frac{-2.43\times10^{-12}m \pm \sqrt{-1.19\times10^{-23}m^2}}{2}\longleftarrow\substack{\text{I get the negative number}\\\text{under the square root}}\\
\end{aligned}