$14.5.24.$ Prove that a free electron can neither absorb nor emit a photon.
### Solution
To prove this, let us write the laws of conservation of energy and momentum for the "electron-photon" system and show that they are satisfied only for a non-existent photon with frequency $\nu = 0$.
<b>1. Impossibility of absorption:</b>
Let us consider the process in a reference frame where the free electron is initially at rest.
- Before absorption: the energy of the electron is $E_{e1} = mc^2$, its momentum is $p_{e1} = 0$. The energy of the incident photon is $E_{\gamma} = h\nu$, its momentum is $p_{\gamma} = \frac{h\nu}{c}$.
- After absorption: the photon disappears. According to the law of conservation of momentum, the entire momentum is transferred to the electron, so $p_{e2} = \frac{h\nu}{c}$. The total energy of the electron becomes $E_{e2} = \sqrt{p_{e2}^2 c^2 + m^2 c^4}$.
Since the electron mass $m \neq 0$, the speed of light $c \neq 0$, and Planck's constant $h \neq 0$, this equality holds true only if **$\nu = 0$**. Photons with zero frequency do not exist; therefore, complete absorption is impossible.
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Since the electron mass $m \neq 0$, the speed of light $c \neq 0$, and Planck's constant $h \neq 0$, this equality holds true only if $\nu = 0$. Photons with zero frequency do not exist; therefore, complete absorption is impossible.
<b>2. Impossibility of emission:</b>
Suppose a free electron at rest emits a photon with frequency $\nu$.
- Before emission: the energy of the system is equal to the rest energy of the electron $E_1 = mc^2$, and the total momentum is zero.
- After emission: a photon appears with momentum $p_{\gamma} = \frac{h\nu}{c}$. To keep the total momentum zero, the electron must acquire a recoil momentum $p_e = -\frac{h\nu}{c}$.
Since $h\nu > 0$, the right side of the equation is always strictly greater than $mc^2$. The equality is possible only if **$\nu = 0$** (meaning no emission occurs).
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Since $h\nu > 0$, the right side of the equation is always strictly greater than $mc^2$. The equality is possible only if $\nu = 0$ (meaning no emission occurs).
#### Answer
The simultaneous satisfaction of the laws of conservation of energy and momentum during the absorption or emission of a photon by a free electron is possible only if the photon frequency is $\nu = 0$. Such photons do not exist.
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#### Answer
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### Statement
### Statement
$14.5.24.$ [Insert the problem statement]
### Solution
### Statement
$14.5.24.$ Prove that a free electron can neither absorb nor emit a photon.
$14.5.24.$ Prove that a free electron can neither absorb nor emit a photon.
### Solution
### Solution
To prove this, let us write the laws of conservation of energy and momentum for the "electron-photon" system and show that they are satisfied only for a non-existent photon with frequency $\nu = 0$.
To prove this, let us write the laws of conservation of energy and momentum for the "electron-photon" system and show that they are satisfied only for a non-existent photon with frequency $\nu = 0$.
<b>1. Impossibility of absorption:</b>
<b>1. Impossibility of absorption:</b>
Let us consider the process in a reference frame where the free electron is initially at rest.
Let us consider the process in a reference frame where the free electron is initially at rest.
- Before absorption: the energy of the electron is $E_{e1} = mc^2$, its momentum is $p_{e1} = 0$. The energy of the incident photon is $E_{\gamma} = h\nu$, its momentum is $p_{\gamma} = \frac{h\nu}{c}$.
- Before absorption: the energy of the electron is $E_{e1} = mc^2$, its momentum is $p_{e1} = 0$. The energy of the incident photon is $E_{\gamma} = h\nu$, its momentum is $p_{\gamma} = \frac{h\nu}{c}$.
- After absorption: the photon disappears. According to the law of conservation of momentum, the entire momentum is transferred to the electron, so $p_{e2} = \frac{h\nu}{c}$. The total energy of the electron becomes $E_{e2} = \sqrt{p_{e2}^2 c^2 + m^2 c^4}$.
- After absorption: the photon disappears. According to the law of conservation of momentum, the entire momentum is transferred to the electron, so $p_{e2} = \frac{h\nu}{c}$. The total energy of the electron becomes $E_{e2} = \sqrt{p_{e2}^2 c^2 + m^2 c^4}$.
Since the electron mass $m \neq 0$, the speed of light $c \neq 0$, and Planck's constant $h \neq 0$, this equality holds true only if **$\nu = 0$**. Photons with zero frequency do not exist; therefore, complete absorption is impossible.
Since the electron mass $m \neq 0$, the speed of light $c \neq 0$, and Planck's constant $h \neq 0$, this equality holds true only if $\nu = 0$. Photons with zero frequency do not exist; therefore, complete absorption is impossible.
<b>2. Impossibility of emission:</b>
<b>2. Impossibility of emission:</b>
Suppose a free electron at rest emits a photon with frequency $\nu$.
Suppose a free electron at rest emits a photon with frequency $\nu$.
- Before emission: the energy of the system is equal to the rest energy of the electron $E_1 = mc^2$, and the total momentum is zero.
- Before emission: the energy of the system is equal to the rest energy of the electron $E_1 = mc^2$, and the total momentum is zero.
- After emission: a photon appears with momentum $p_{\gamma} = \frac{h\nu}{c}$. To keep the total momentum zero, the electron must acquire a recoil momentum $p_e = -\frac{h\nu}{c}$.
- After emission: a photon appears with momentum $p_{\gamma} = \frac{h\nu}{c}$. To keep the total momentum zero, the electron must acquire a recoil momentum $p_e = -\frac{h\nu}{c}$.
Since $h\nu > 0$, the right side of the equation is always strictly greater than $mc^2$. The equality is possible only if **$\nu = 0$** (meaning no emission occurs).
Since $h\nu > 0$, the right side of the equation is always strictly greater than $mc^2$. The equality is possible only if $\nu = 0$ (meaning no emission occurs).
#### Answer
#### Answer
The simultaneous satisfaction of the laws of conservation of energy and momentum during the absorption or emission of a photon by a free electron is possible only if the photon frequency is $\nu = 0$. Such photons do not exist.
The simultaneous satisfaction of the laws of conservation of energy and momentum during the absorption or emission of a photon by a free electron is possible only if the photon frequency is $\nu = 0$. Such photons do not exist.
