A particle of mass $m$ in an infinite potential well of width $a$ is subjected to a perturbation, $V' = \frac{h^2}{40ma^2}$ as shown in figure, where $h$ is Planck's constant. 
The first order energy shift of the fourth energy eigenstate due to this perturbation is
$(\frac{h^2}{Nma^2})$
The value of $N$ is ____________ (in integer).
1. Identify the wavefunction for the $n$-th energy state:
For an infinite potential well of width $a$ defined from $x = 0$ to $x = a$, the normalized wavefunctions are given by:
$\psi_n(x) = \sqrt{\frac{2}{a}} \sin\left(\frac{n\pi x}{a}\right)$
For the fourth energy eigenstate ($n = 4$):
$\psi_4(x) = \sqrt{\frac{2}{a}} \sin\left(\frac{4\pi x}{a}\right)$
2. Set up the first-order energy shift formula:
According to first-order perturbation theory, the energy shift $E_n^{(1)}$ is the expectation value of the perturbation $V'$ in the unperturbed state:
$E_n^{(1)} = \int_0^a \psi_n^*(x) V'(x) \psi_n(x) \, dx$
From the figure, $V'(x)$ is non-zero only in the region from $x = \frac{3a}{4}$ to $x = a$:
$E_4^{(1)} = \int_{3a/4}^a \left( \sqrt{\frac{2}{a}} \sin\left(\frac{4\pi x}{a}\right) \right)^2 \cdot V' \, dx$
$E_4^{(1)} = \frac{2V'}{a} \int_{3a/4}^a \sin^2\left(\frac{4\pi x}{a}\right) \, dx$
3. Evaluate the integral:
Use the trigonometric identity $\sin^2\theta = \frac{1 - \cos 2\theta}{2}$:
$\int_{3a/4}^a \sin^2\left(\frac{4\pi x}{a}\right) \, dx = \int_{3a/4}^a \frac{1 - \cos(8\pi x/a)}{2} \, dx$
$= \frac{1}{2} \left[ x - \frac{a}{8\pi} \sin\left(\frac{8\pi x}{a}\right) \right]_{3a/4}^a$
$= \frac{1}{2} \left[ \left( a - 0 \right) - \left( \frac{3a}{4} - \frac{a}{8\pi} \sin(6\pi) \right) \right]$
$= \frac{1}{2} \left[ a - \frac{3a}{4} \right] = \frac{1}{2} \left( \frac{a}{4} \right) = \frac{a}{8}$
4. Calculate $E_4^{(1)}$ and find N:
Substitute the value of the integral and $V' = \frac{h^2}{40ma^2}$ back into the equation:
$E_4^{(1)} = \frac{2}{a} \cdot \left( \frac{h^2}{40ma^2} \right) \cdot \frac{a}{8}$
$E_4^{(1)} = \frac{2h^2}{320ma^2} = \frac{h^2}{160ma^2}$
Comparing this with the given form $\frac{h^2}{Nma^2}$:
$N = 160$
The value of N is 160.
Important Questions from Perturbation Theory Time Independent Degenerate
- If the perturbation $V = \lambda x^3$ is added to the Hamiltonian of a one-dimensional harmonic oscillator, the matrix element $\langle m|V|0 \rangle$ is/are non-zero for which of the following states? Here, the eigenstates of the harmonic oscillator are denoted by $|n\rangle$.
A two-level quantum system has energy eigenvalues $E_1$ and $E_2$. A perturbing potential $H' = \lambda \Delta \sigma_x$ is introduced, where $\Delta$ is a constant having dimensions of energy, $\lambda$ is a small dimensionless parameter, and $\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$. The magnitudes of the first and the second order corrections to $E_1$ due to $H'$, respectively, are
The ground state energy of a particle of mass $m$ in an infinite potential well is $E_0$. It changes to $E_0(1 + \alpha \times 10^{-3})$, when there is a small potential bump of height $V_0 = \frac{\pi^2 \hbar^2}{50mL^2}$ and width $a = L/100$, as shown in the figure. The value of $\alpha$ is ________ (up to two decimal places).
- A particle of mass $m$ in the x-y plane is confined in an infinite two-dimensional well with vertices at $(0, 0)$, $(0, L)$, $(L, L)$, $(L, 0)$. The eigenfunctions of this particle are $\Psi_{n_x,n_y} = \sin(\frac{n_x\pi x}{L}) \sin(\frac{n_y\pi y}{L})$. If perturbation of the form $V = Cxy$, where $C$ is a real constant, is applied, then which of the following statements are correct for the first excited state?
Consider the Hamiltonian $\hat{H} = \hat{H}_0 + \hat{H}'$ where
\[\hat{H}_0 = \begin{pmatrix} E & 0 & 0 \\ 0 & E & 0 \\ 0 & 0 & E \end{pmatrix}\]and $\hat{H}$ is the time independent perturbation given by
\[\hat{H}' = \begin{pmatrix} 0 & k & 0 \\ k & 0 & k \\ 0 & k & 0 \end{pmatrix}\]where $k>0$. If, the maximum energy eigenvalue of $\hat{H}$ is 3 eV corresponding to $E=2$ eV, the value of $k$ (rounded off to three decimal places) in eV is ________.
