Energy-Time Uncertainty Principle
The natural line width of a spectral line is determined by the finite lifetime of the excited atomic state. This relationship is governed by the Heisenberg uncertainty principle, which states that the product of the uncertainty in energy ($\Delta E$) and the uncertainty in time ($\Delta t$) is at least proportional to the reduced Planck constant ($\hbar$):
$ \Delta E \Delta t \gtrsim \frac{\hbar}{2} $
In this context, the lifetime of the state ($\tau$) serves as the time uncertainty ($\Delta t$). Therefore, the natural line width (minimum energy uncertainty $\Delta E$) can be approximated as:
$ \Delta E \approx \frac{\hbar}{\tau} $
Natural Line Width Calculation
We are given:
- Lifetime of the atomic state, $\tau = 1 \text{ nanosecond} = 1 \times 10^{-9} \text{ s}$.
- The reduced Planck constant, $\hbar \approx 6.58 \times 10^{-16} \text{ eV·s}$.
Applying the formula:
$ \Delta E \approx \frac{\hbar}{\tau} $
Substituting the values:
$ \Delta E \approx \frac{6.58 \times 10^{-16} \text{ eV·s}}{1 \times 10^{-9} \text{ s}} $
Calculating the energy width:
$ \Delta E \approx 6.58 \times 10^{-7} \text{ eV} $
Order of Magnitude Determination
The calculated natural line width is approximately $6.58 \times 10^{-7}$ eV. We must select the option that best represents the order of magnitude of this result.
The available options are:
- $10^{-10}$ eV
- $10^{-9}$ eV
- $10^{-8}$ eV
- $10^{-4}$ eV
Comparing our calculated value ($6.58 \times 10^{-7}$ eV) to the options, it is found to be closest in order of magnitude to $10^{-8}$ eV.
Important Questions from Operators Commutators Heisenberg Picture
- Consider an operator $\hat{A}$ which is not Hermitian. Find the possible values of $c$ and $d$ such that the operator $(c\hat{A} - d\hat{A}^\dagger)$ is Hermitian.
- Which of the following operators is/are self-adjoint?
- Consider operators $\hat{A}$, $\hat{B}$, and $\hat{C}$ for three observables of a quantum system satisfying $[\hat{A}, \hat{B}] = 0$, $[\hat{B}, \hat{C}] = 0$, and $[\hat{A}, \hat{C}] \neq 0$, with uncertainties $\Delta A, \Delta B, \Delta C$, respectively. From the options given below, which is/are implied by the commutation relations among $\hat{A}, \hat{B}$, and $\hat{C}$?
- Let $|m\rangle$ and $|n\rangle$ denote the energy eigenstates of a one-dimensional simple harmonic oscillator. The position and momentum operators are $\hat{X}$ and $\hat{P}$, respectively. The matrix element $\langle m|\hat{P}\hat{X}|n\rangle$ is non-zero when
The wavefunction of a particle in one dimension is given by
$\psi(x) = \begin{cases} M, & -a < x < a \\ 0, & \text{otherwise.} \end{cases}$
Here $M$ and $a$ are positive constants. If $\phi(p)$ is the corresponding momentum space wavefunction, which one of the following plots best represents $|\phi(p)|^2$ ?
