Estimating Minimum Potential Well Depth

The problem asks for the minimum depth of a one-dimensional potential well required to have at least one bound state for an electron. We can estimate this using the Heisenberg uncertainty principle, which relates the uncertainty in position ($\Delta x$) and the uncertainty in momentum ($\Delta p$):

$ \Delta x \cdot \Delta p \ge \frac{\hbar}{2} $

Here, $\hbar$ is the reduced Planck constant, $\hbar = h / (2\pi)$.

Applying the Uncertainty Principle

We are given:

• Well width, $\Delta x \approx L = 3 \text{ nm} = 3 \times 10^{-9} \text{ m}$.
• Electron mass, $m_e = 9.31 \times 10^{-31} \text{ kg}$.
• Planck constant, $h = 6.626 \times 10^{-34} \text{ J s}$.
• Elementary charge, $e = 1.602 \times 10^{-19} \text{ C}$.

First, calculate the reduced Planck constant $\hbar$:

$ \hbar = \frac{h}{2\pi} = \frac{6.626 \times 10^{-34} \text{ J s}}{2\pi} \approx 1.054 \times 10^{-34} \text{ J s} $

Now, estimate the minimum uncertainty in momentum ($\Delta p$) using the uncertainty principle:

$ \Delta p \ge \frac{\hbar}{2 \Delta x} = \frac{1.054 \times 10^{-34} \text{ J s}}{2 \times (3 \times 10^{-9} \text{ m})} \approx 1.757 \times 10^{-26} \text{ kg m/s} $

Calculating Minimum Kinetic Energy

The minimum kinetic energy ($E_{min}$) of the electron confined within the well can be estimated using the uncertainty in momentum. The relationship between kinetic energy and momentum is $E = p^2 / (2m)$. We use $\Delta p$ as a proxy for the momentum magnitude:

$ E_{min} \approx \frac{(\Delta p)^2}{2 m_e} $

$ E_{min} \approx \frac{(1.757 \times 10^{-26} \text{ kg m/s})^2}{2 \times (9.31 \times 10^{-31} \text{ kg})} $

$ E_{min} \approx \frac{3.087 \times 10^{-52} \text{ kg}^2 \text{ m}^2/\text{s}^2}{1.862 \times 10^{-30} \text{ kg}} \approx 1.658 \times 10^{-22} \text{ J} $

Converting Energy to Electronvolts (eV)

To compare with the options, convert the minimum energy from Joules to electronvolts (eV) using the conversion factor $1 \text{ eV} = 1.602 \times 10^{-19} \text{ J}$:

$ E_{min} (\text{eV}) = \frac{1.658 \times 10^{-22} \text{ J}}{1.602 \times 10^{-19} \text{ J/eV}} \approx 1.035 \times 10^{-3} \text{ eV} $

This value is approximately $1.035$ millielectronvolts (meV).

Minimum Well Depth Requirement

For at least one bound state to exist, the potential well must be deep enough to contain the electron's minimum kinetic energy. Therefore, the minimum depth ($V_0$) of the well must be at least this estimated minimum kinetic energy:

$ V_0 \approx E_{min} \approx 1.035 \text{ meV} $

This value is closest to $1 \text{ meV}$.