Answer to Question #169200 in Quantum Mechanics for Mohammed Ismail

Question

 Three spin less non-interacting distinguishable particles, with respective masses  𝑚1, 𝑚2, and 𝑚3 in the ratio 𝑚1: 𝑚2: 𝑚3 = 1: 2: 3, are subject to a common infinite  square well potential of width L in one spatial dimension. Determine the energies and the corresponding wave functions in the lowest two energy states of the system.

Answer 1

Answer

common infinite square well potential of width L in one spatial dimension

Energy is given

$E_n=\frac{n^2\pi^2\hbar^2}{2mL^2}$

Wavefunction

$\psi=\sqrt{\frac{2}{L}}sin(\frac{n\pi x}{L})$

According to given data total energy of system is given

For ground state n=1

$E_1=\frac{n^2\pi^2\hbar^2}{2m_1L^2}+\frac{n^2\pi^2\hbar^2}{2m_2L^2}+\frac{n^2\pi^2\hbar^2}{2m_3L^2}$

$E_1=$ $\frac{7\pi^2\hbar^2}{8mL^2}$

First excited state n=2

$E_2=\frac{n^2\pi^2\hbar^2}{2m_1L^2}+\frac{n^2\pi^2\hbar^2}{2m_2L^2}+\frac{n^2\pi^2\hbar^2}{2m_3L^2}$

=$E_1=\frac{7\pi^2\hbar^2}{2mL^2}$

Now wavefunction

$\psi_1=\sqrt{\frac{2}{L}}sin(\frac{\pi x}{L})$

And

$\psi_2=\sqrt{\frac{2}{L}}sin(\frac{2\pi x}{L})$