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
Quantum Mechanics: Non-Interacting Particles in 1D Box
System Description
The problem involves two non-interacting particles confined within a one-dimensional box with infinite potential barriers. We are given their individual wavefunctions ($\psi_1, \psi_2$) and energy levels ($E_1, E_2$). The task is to identify the incorrect statements about this system.
Principles for Non-Interacting Particles
For systems composed of non-interacting particles:
- Total Energy: The total energy of the system is the simple sum of the individual energies. If the particles have energies $E_1$ and $E_2$, the total energy $E_{total}$ is:
$ E_{total} = E_1 + E_2 $
- Total Wavefunction: The total wavefunction is generally the product of the individual wavefunctions. For particle 1 in state $\psi_1$ and particle 2 in state $\psi_2$, the combined wavefunction $\Psi(x_1, x_2)$ is:
$ \Psi(x_1, x_2) = \psi_1(x_1) \psi_2(x_2) $
(Note: For identical particles, this wavefunction must be symmetrized or antisymmetrized.)
Statement Evaluation
- Statement 1: The total energy is $E_1 + E_2$.
This is correct based on the principle of energy addition for non-interacting particles. - Statement 2: The total wavefunction is $\psi_1 + \psi_2$.
This is incorrect. Wavefunctions for distinct non-interacting particles combine via multiplication, not addition. Addition describes superposition for a single particle's states. - Statement 3: The total energy is $E_1E_2$.
This is incorrect. The total energy is the sum ($E_1 + E_2$), not the product ($E_1E_2$). - Statement 4: The total wavefunction is $\psi_1\psi_2$.
This is correct. It represents the product wavefunction for two distinguishable, non-interacting particles.
Conclusion
The question asks for the incorrect statements. Based on the analysis:
- Statement 2 is incorrect because wavefunctions add for superposition states of one particle, not combination of two particles.
- Statement 3 is incorrect because energies add for non-interacting particles, they do not multiply.
Therefore, the incorrect statements correspond to options B and C.
Important Questions from Particle in a Box
- 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)- The wavelength associated with a particle in one-dimensional box of length $L$ is ($n$ refers to the quantum number)
- 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 $m_e =9.1\times10^{-31} \text{ kg}$, $h=6.6\times10^{-34} \text{ Js}$, $c=3.0\times10^8 \text{ m s}^{-1}$)
