Particle Wavelength in 1D Box
For a particle confined within a one-dimensional box of length $L$, the wave nature of the particle leads to quantized states. The allowed wavelengths must form standing waves within the box.
Deriving the Wavelength Formula
The condition for standing waves in a 1D box requires that the length of the box $L$ must accommodate an integer number ($n$) of half-wavelengths ($\lambda_n / 2$). Mathematically, this is expressed as:
$ L = n \frac{\lambda_n}{2} $
Where:
- $L$ is the length of the box.
- $n$ is the quantum number ($n = 1, 2, 3, ...$).
- $\lambda_n$ is the wavelength associated with the quantum state $n$.
Calculating Wavelength
To find the wavelength $\lambda_n$, we rearrange the formula:
$ \lambda_n = \frac{2L}{n} $
This formula gives the wavelength of the particle corresponding to each allowed quantum state $n$. For the ground state ($n=1$), the wavelength is $2L$. As the quantum number $n$ increases, the wavelength decreases.
Conclusion
The wavelength associated with a particle in a one-dimensional box of length $L$ is given by the formula $\frac{2L}{n}$. This matches Option A.
Important Questions from Particle in a Box
Consider two non-interacting particles confined to a one-dimensional box with infinite potential barriers. Their wavefunctions are $\psi_1$ and $\psi_2$ and energies are $E_1$ and $E_2$, respectively. The INCORRECT statement(s) about this system is/are
- Wavefunctions and energies for a particle confined in a cubic box are $\psi_{n_x,n_y,n_z}$ and $E_{n_x,n_y,n_z}$, respectively. The functions $\Phi_1$, $\Phi_2$, $\Phi_3$, and $\Phi_4$ are written as linear combinations of $\psi_{n_x,n_y,n_z}$. Among these functions, the eigenfunction(s) of the Hamiltonian operator for this particle is/are
$\Phi_1 = \frac{1}{\sqrt{2}}\psi_{1,4,1} - \frac{1}{\sqrt{2}}\psi_{2,2,3}$
$\Phi_2 = \frac{1}{\sqrt{2}}\psi_{1,5,1} + \frac{1}{\sqrt{2}}\psi_{3,3,3}$
$\Phi_3 = \frac{1}{\sqrt{2}}\psi_{1,3,8} + \frac{1}{\sqrt{2}}\psi_{3,8,1}$
$\Phi_4 = \frac{1}{2}\psi_{3,3,1} + \frac{\sqrt{3}}{2}\psi_{2,4,1}$ The wave function of a particle in a cubic box (of side L) is given by
$\psi(x, y, z) = \sqrt{32/L^3} \sin \frac{\pi x}{L} \cos \frac{\pi x}{L} \sin \frac{2\pi y}{L} \sin \frac{\pi z}{L}$.
The ratio of the energy of the state corresponding to the above wave function to the ground state energy is ________.
(rounded off to the nearest integer)- Assume 1,3,5-hexatriene to be a linear molecule and model the $\pi$ electrons as particles in a one-dimensional box of length 0.70 nm. The wavelength (in nm) corresponding to the transition from the ground-state to the first excited-state is ________
The $\pi$ electrons in benzene can be modelled as particles in a ring that follow Pauli's exclusion principle. Given that the radius of benzene is 1.4 Å, the longest wavelength of light that is absorbed during an electronic transition in benzene is ____________ nm. (Up to one decimal place. Use $m_e =9.1\times10^{-31} \text{ kg}$, $h=6.6\times10^{-34} \text{ Js}$, $c=3.0\times10^8 \text{ m s}^{-1}$)
