Question

The system with the lowest zero-point energy when it is confined to a one dimensional box of length 𝐿 is

• A. an electron
• B. a hydrogen atom
• C. a helium atom
• D. a proton

Answer 1

The Correct Option is C

To determine which system has the lowest zero-point energy when confined to a one-dimensional box of length \( L \), we need to consider the physics of a particle in a box. According to quantum mechanics, the zero-point energy \( E_1 \) for a particle in a one-dimensional box of length \( L \) is given by:

\(E_1 = \frac{h^2}{8mL^2}\) 

Where:

• \(h\) is Planck's constant.
• \(m\) is the mass of the particle.
• \(L\) is the length of the box.

From the formula, it is clear that the zero-point energy is inversely proportional to the mass of the particle \( m \). Therefore, the larger the mass, the lower the zero-point energy.

Let's compare the mass of different particles given in the options:

• Mass of electron: \(9.11 \times 10^{-31} \, \text{kg}\)
• Mass of proton: \(1.67 \times 10^{-27} \, \text{kg}\)
• Mass of hydrogen atom: approximately \(1.67 \times 10^{-27} \, \text{kg}\)
• Mass of helium atom: \(4.00 \times 1.67 \times 10^{-27} \, \text{kg}\) (since it has two protons and two neutrons)

Among these, the helium atom has the largest mass. Therefore, it will have the lowest zero-point energy because the zero-point energy decreases with an increase in the particle's mass.

Therefore, the correct answer is: a helium atom.