Question

A beam of electrons moving in the positive \(x\) -direction encounters a potential barrier that is \(2.51 \mathrm{eV}\) high and \(1.00 \mathrm{nm}\) wide. Each electron has a kinetic energy of \(2.50 \mathrm{eV},\) and the electrons arrive at the barrier at a rate of \(1000 .\) electrons/s. What is the rate \(I_{T}\) (in electrons/s) at which electrons pass through the barrier, on average? What is the rate \(I_{\mathrm{R}}\) (in electrons/s) at which electrons bounce back from the barrier, on average? Determine and compare the wavelengths of the electrons before and after they pass through the barrier.

Short answer

Answer: The transmission rate and reflection rate of electrons passing through a potential barrier can be calculated using the transmission probability (T) and reflection probability (R), which can be found using the barrier penetration model. To compare the wavelength of electrons before and after passing through the barrier, the de Broglie wavelength equation is used, taking into account the change in kinetic energy as they interact with the barrier. Due to the quantum tunneling phenomenon, a significant change in electron wavelength may occur as they pass through the barrier.

Step-by-step solution

Calculate the transmission probability and reflection probability

Firstly, we need to calculate the transmission probability (\(T\)) and reflection probability (\(R\)) of the electrons at the barrier using the barrier penetration model. From quantum mechanics, the transmission probability can be calculated by: \(T = \exp(-2\kappa L)\) Where \(\kappa =\dfrac{\sqrt{2mo(V-E)}}{\hbar}\), \(mo\) is the electron's mass, \(V\) is the potential barrier height, \(E\) is the kinetic energy of the electron, \(L\) is the barrier width, and \(\hbar\) is the reduced Planck's constant. Now, we can calculate the reflection probability as \(R = 1 - T\).

Calculate the rate at which electrons pass through and bounce back from the barrier

Now that we have the transmission probability (\(T\)) and reflection probability (\(R\)), we can find the transmission rate (\(I_T\)) and reflection rate (\(I_R\)). Since the electrons arrive at the barrier with a rate of \(1000\) electrons/s, we can say: \(I_T = T * 1000\) electrons/s \(I_R = R * 1000\) electrons/s

Calculate the wavelength of electrons before and after passing through the barrier

To compare the wavelengths of electrons before and after passing through the barrier, we will use the de Broglie wavelength equation: \(\lambda = \dfrac{h}{p}\) Where \(\lambda\) is the wavelength, \(h\) is the Planck's constant, and \(p\) is the momentum of the electron. Before interacting with the barrier, the electron has a kinetic energy of \(2.50 eV\), which should be converted to Joules as \(E_j = E_ev * e\) (where \(e\) is the elementary charge). Then, we can calculate the momentum \(p\) using the relation: \(p =\sqrt{2m_oE_j}\) Therefore, we can now calculate the wavelength before interaction, \(\lambda_i\), as: \(\lambda_i = \dfrac{h}{\sqrt{2moE_{ji}}}\) After passing the barrier, the electron has a kinetic energy of \(2.50eV - 2.51eV = -0.01eV\). Convert this to Joules as \(E_j = E_ev * e\), and calculate the momentum \(p\) using the same relation as before. Now, we can calculate the wavelength after interaction, \(\lambda_f\), as: \(\lambda_f = \dfrac{h}{\sqrt{2moE_{jf}}}\) Finally, we can compare the wavelengths \(\lambda_i\) and \(\lambda_f\). Note that since the electrons pass through the barrier, their kinetic energy decreases (negative value), and the denominator of the de Broglie wavelength equation becomes imaginary. Therefore, a significant change in electron wavelength may occur, which is a result of the quantum tunneling phenomenon.