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?
Potential Barrier Wavefunction Analysis
The problem asks for a valid quantum mechanical wavefunction ($\Psi$) for a particle in a specified potential $V(x)$, given that the particle's energy $E$ is less than the barrier height $V_0$ ($E < V_0$). The potential is defined as:
$ V(x) = \begin{cases} \infty, & \text{if } x \le 0 \\ V_0, & \text{if } a \le x \le b \\ 0, & \text{otherwise} \end{cases} $
We need to analyze the wavefunction's behavior in each region based on the potential.
Wave Behavior Across Potential Regions
- Region $x \le 0$: The potential $V(x)$ is infinite. The wavefunction must be zero, $\Psi(x) = 0$, because a particle cannot exist in a region of infinite potential.
- Region $0 < x < a$: The potential $V(x) = 0$. The time-independent Schrödinger equation is $-\frac{\hbar^2}{2m}\frac{d^2\Psi}{dx^2} = E\Psi$. The solutions are oscillatory, represented by functions like $e^{\pm ikx}$, where $k = \sqrt{2mE}/\hbar$.
- Region $a \le x \le b$: This is a finite potential barrier where $V(x) = V_0$. Since the particle's energy $E < V_0$, the Schrödinger equation is $-\frac{\hbar^2}{2m}\frac{d^2\Psi}{dx^2} + V_0\Psi = E\Psi$. This rearranges to $\frac{d^2\Psi}{dx^2} = \frac{2m(V_0 - E)}{\hbar^2}\Psi$. Let $\kappa^2 = \frac{2m(V_0 - E)}{\hbar^2}$. The solutions are real exponentials, $\Psi(x) \propto e^{\pm \kappa x}$. Physically, this means the wavefunction decays exponentially within this barrier region, typically represented by $e^{-\kappa x}$ from the side where the wave enters.
- Region $x > b$: Similar to the region $0 < x < a$, the potential $V(x) = 0$. The wavefunction is oscillatory, $\Psi(x) \propto e^{\pm ikx}$.
Valid Wavefunction Conditions
A physically valid wavefunction for this system must satisfy:
- It must be zero in the region of infinite potential ($\Psi(x \le 0) = 0$).
- It must be continuous at the boundaries between regions ($x=a$ and $x=b$). This applies to both $\Psi(x)$ and its derivative $\Psi'(x)$.
- It must show exponential decay (not oscillation) within the finite potential barrier where $E < V_0$ (region $a \le x \le b$).
- It must be oscillatory in the regions where the potential is zero ($0 < x < a$ and $x > b$).
Selecting the Correct Wavefunction Schematic
Evaluating the options based on these conditions:
- Option 1 incorrectly shows oscillatory behavior within the barrier region.
- Option 2 correctly represents the wavefunction: it is zero for $x \le 0$, shows oscillatory behavior in the free regions ($0 < x < a$ and $x > b$), and importantly, exhibits exponential decay within the barrier ($a \le x \le b$) consistent with $E < V_0$. The wavefunction is continuous at the boundaries.
- Option 3 incorrectly shows a zero wavefunction within the barrier.
- Option 4 incorrectly shows exponential growth within the barrier, which is not physically valid for typical bound or transmitted states in this scenario.
Therefore, the schematic in Option 2 is the correct representation of the quantum mechanical wavefunction.
Important Questions from Schrödinger Equation 1D Potentials Harmonic Oscillator
- The energy $E$ and degeneracy $d$ of the second excited state of a three-dimensional, isotropic quantum harmonic oscillator with angular frequency $\omega$ are
- A particle of mass $ m $ is in a potential $ V(x) = \frac{1}{2}m\omega^2x^2 $ for $ x > 0 $ and $ V(x) = \infty $ for $ x \leq 0 $, where $ \omega $ is the angular frequency. The ratio of the energies corresponding to the lowest energy level to the next higher level is
- 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?
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
