The wavefunction of a particle in an infinite one-dimensional potential well at time $t$ is
$\Psi(x, t) = \sqrt{\frac{2}{3}} e^{-iE_1t/\hbar}\psi_1(x) + \frac{1}{\sqrt{6}} e^{i\pi/6}e^{-iE_2t/\hbar}\psi_2(x) + \frac{1}{\sqrt{6}} e^{i\pi/4}e^{-iE_3t/\hbar}\psi_3(x)$
where $\psi_1, \psi_2$ and $\psi_3$ are the normalized ground state, the normalized first excited state and the normalized second excited state, respectively. $E_1, E_2$ and $E_3$ are the eigen-energies corresponding to $\psi_1, \psi_2$ and $\psi_3$, respectively. The expectation value of energy of the particle in state $\Psi(x, t)$ is
To find the expectation value of energy, $\langle E \rangle$, for a particle in a given quantum state $\Psi(x, t)$, we use the formula:
$ \langle E \rangle = \sum_n |c_n|^2 E_n $
where $c_n$ are the probability amplitudes and $E_n$ are the corresponding eigen-energies.
Deriving Expectation Value Components
The given wavefunction is:
$ \Psi(x, t) = \sqrt{\frac{2}{3}} e^{-iE_1t/\hbar}\psi_1(x) + \frac{1}{\sqrt{6}} e^{i\pi/6}e^{-iE_2t/\hbar}\psi_2(x) + \frac{1}{\sqrt{6}} e^{i\pi/4}e^{-iE_3t/\hbar}\psi_3(x) $
The coefficients ($c_n$) and their squared magnitudes ($|c_n|^2$) are:
- For $n=1$: $c_1 = \sqrt{\frac{2}{3}}$. So, $|c_1|^2 = \left(\sqrt{\frac{2}{3}}\right)^2 = \frac{2}{3}$.
- For $n=2$: $c_2 = \frac{1}{\sqrt{6}} e^{i\pi/6}$. So, $|c_2|^2 = \left|\frac{1}{\sqrt{6}} e^{i\pi/6}\right|^2 = \left(\frac{1}{\sqrt{6}}\right)^2 |e^{i\pi/6}|^2 = \frac{1}{6} \times 1 = \frac{1}{6}$.
- For $n=3$: $c_3 = \frac{1}{\sqrt{6}} e^{i\pi/4}$. So, $|c_3|^2 = \left|\frac{1}{\sqrt{6}} e^{i\pi/4}\right|^2 = \left(\frac{1}{\sqrt{6}}\right)^2 |e^{i\pi/4}|^2 = \frac{1}{6} \times 1 = \frac{1}{6}$.
Calculating Total Expectation Energy
For a particle in an infinite one-dimensional potential well, the energy levels are proportional to the square of the state number ($n$). Thus, $E_n \propto n^2$. We can write this relationship relative to the ground state energy $E_1$:
$ E_2 = 2^2 E_1 = 4E_1 $ $ E_3 = 3^2 E_1 = 9E_1 $
Now substitute the squared coefficients and the energy relations into the expectation value formula:
$ \langle E \rangle = |c_1|^2 E_1 + |c_2|^2 E_2 + |c_3|^2 E_3 $ $ \langle E \rangle = \left(\frac{2}{3}\right) E_1 + \left(\frac{1}{6}\right) (4E_1) + \left(\frac{1}{6}\right) (9E_1) $
Simplify the expression:
$ \langle E \rangle = \frac{2}{3} E_1 + \frac{4}{6} E_1 + \frac{9}{6} E_1 $ $ \langle E \rangle = \frac{4}{6} E_1 + \frac{4}{6} E_1 + \frac{9}{6} E_1 $ $ \langle E \rangle = \frac{4 + 4 + 9}{6} E_1 $ $ \langle E \rangle = \frac{17}{6} E_1 $
The expectation value of the energy is $\frac{17}{6}E_1$.
Important Questions from Schrödinger Equation 1D Potentials Harmonic Oscillator
A particle is subjected to a potential
$V(x) = \begin{cases} \infty, & x \le 0 \\ V_0, & a \le x \le b \\ 0, & \text{elsewhere} \end{cases}$
Here, $a > 0$ and $b > a$. If the energy of the particle $E < V_0$, which one of the following schematics is a valid quantum mechanical wavefunction ($\Psi$) for the system?- The wavefunction for a particle is given by the form $e^{-(iax+\beta)}$, where $a$ and $\beta$ are real constants. In which one of the following potentials $V(x)$, the particle is moving?
A particle of mass $m$ is moving in the potential
$V(x) = \begin{cases} V_0 + \frac{1}{2}m\omega_0^2x^2, & x > 0, \\ \infty, & x \le 0, \end{cases}$
Figures P, Q, R and S show different combinations of the values of $\omega_0$ and $V_0$.
$E_j^{(P)}, E_j^{(Q)}, E_j^{(R)}$ and $E_j^{(S)}$ with $j = 0, 1, 2, ...$, are the eigen-energies of the $j$-th level for the potentials shown in figures P, Q, R and S, respectively. Which of the statement is/are true?- Young's double slit experiment is performed using a beam of $C_{60}$ (fullerene) molecules, each molecule being made up of 60 carbon atoms. When the slit separation is 50 nm, fringes are formed on a screen kept at a distance of 1 m from the slits. Now, the experiment is repeated with $C_{70}$ molecules with a slit separation of 92.5 nm. The kinetic energies of both the beams are the same. The position of the 4th bright fringe for $C_{60}$ will correspond to the $n^{th}$ bright fringe for $C_{70}$. What is the value of $n$ (rounded off to the nearest integer) ?
- Consider a particle in a two dimensional infinite square well potential of side $L$, with $0 \le x \le L$ and $0 \le y \le L$. The wavefunction of the particle is zero only along the line $y = \frac{L}{2}$, apart from the boundaries of the well. If the energy of the particle in this state is $E$, what is the energy of the ground state?
