A particle is constrained to move in a truncated harmonic potential well ($x > 0$) as shown in the figure. Which one of the following statements is CORRECT?
In this problem, we are dealing with a particle constrained to move in a truncated harmonic potential well, where \( x > 0 \). This means that the potential energy function \( V(x) \) is only defined for \( x > 0 \) and is harmonic in nature.
The standard quantum harmonic oscillator potential is given by:
\(V(x) = \frac{1}{2}m\omega^2x^2\)
The energy levels of a typical quantum harmonic oscillator are:
\(E_n = \left(n + \frac{1}{2}\right)\hbar\omega\) where \( n = 0, 1, 2, \ldots \)
Now, consider the conditions given in the problem:
- The potential is only defined for \( x > 0 \), implying that the parity symmetry present in a full harmonic potential is broken.
Let's evaluate the given options:
- The parity of the first excited state is even: In a truncated potential like this, parity is not well-defined because the wave functions do not extend over \( x < 0 \).
- The parity of the ground state is even: Similarly, the notion of parity is not applicable here due to the restriction to \( x > 0 \).
- The ground state energy is \(\frac{1}{2}\hbar\omega\): For a half harmonic oscillator starting from \( x > 0 \), the energies are shifted. The energy levels are typically higher than the standard oscillator.
- The first excited state energy is \(\frac{7}{2}\hbar\omega\): This is correct for a particle in a half-space harmonic oscillator potential, where energy levels are adjusted because the node structure and allowed states change. This matches the known result for the first excited state of a half-harmonic oscillator.
Thus, the correct answer is that the first excited state energy is \(\frac{7}{2}\hbar\omega\).
Important Questions from Schrödinger Equation 1D Potentials Harmonic Oscillator
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 $\Psi(x, t)$ isA 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) ?
