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Psi-String
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Problem:An electron flies toward +x direction with velocity of 0.9c, while a positron flies toward -y direction with the same velocity. Assuming their speed is so fast that they collide and annihilate at the origin, what will be the magnitude and the direction of the wave vector of the generated photon, k?
I have two kinds of solution. One is base on conservation of linear momentum, one is base on conservation of energy, but they don't have the same result! Where went wrong??
Method 1
By conservation of momentum,
[tex] P_i = \frac{m_e v}{\sqrt{1-v^2/c^2}} \hat{i} - \frac{m_e v}{\sqrt{1-v^2/c^2}} \hat{j}[/tex]
since the magnitude is the same in the two direction, we have
[tex] P_{photon} \cos \frac{\pi}{4} = \frac{m_e v}{\sqrt{1-v^2/c^2}} [/tex]
[tex] p = \gamma m_ev \sqrt{2} = \frac{h}{\lambda} [/tex]
so [tex] k = \frac{2\sqrt{2}\pi m_e v}{h\sqrt{1-v^2/c^2}}[/tex]
substitute v=0.9c
[tex] k = \frac{1.8\sqrt{2} \pi m_e c}{h \sqrt{1-(0.9)^2}}[/tex]
Method 2
By conservation of energy
[tex] 2 \times \gamma m c^2 = \frac{hc}{\lambda} [/tex]
then
[tex] k = \frac{2 \pi}{\lambda} = \frac{4 m_e c \pi}{h\sqrt{1-v^2/c^2}} = \frac{4 m_e c \pi}{h \sqrt{1-0.9^2}}[/tex]
why the two method turn out different result??
thanks for help!
I have two kinds of solution. One is base on conservation of linear momentum, one is base on conservation of energy, but they don't have the same result! Where went wrong??
Method 1
By conservation of momentum,
[tex] P_i = \frac{m_e v}{\sqrt{1-v^2/c^2}} \hat{i} - \frac{m_e v}{\sqrt{1-v^2/c^2}} \hat{j}[/tex]
since the magnitude is the same in the two direction, we have
[tex] P_{photon} \cos \frac{\pi}{4} = \frac{m_e v}{\sqrt{1-v^2/c^2}} [/tex]
[tex] p = \gamma m_ev \sqrt{2} = \frac{h}{\lambda} [/tex]
so [tex] k = \frac{2\sqrt{2}\pi m_e v}{h\sqrt{1-v^2/c^2}}[/tex]
substitute v=0.9c
[tex] k = \frac{1.8\sqrt{2} \pi m_e c}{h \sqrt{1-(0.9)^2}}[/tex]
Method 2
By conservation of energy
[tex] 2 \times \gamma m c^2 = \frac{hc}{\lambda} [/tex]
then
[tex] k = \frac{2 \pi}{\lambda} = \frac{4 m_e c \pi}{h\sqrt{1-v^2/c^2}} = \frac{4 m_e c \pi}{h \sqrt{1-0.9^2}}[/tex]
why the two method turn out different result??
thanks for help!
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