The Quantum Oscilloscope Probe · AI Generator #GEN-DDDA6782

Physics · quantum-mechanics

Problem

A cutting-edge quantum oscilloscope uses a probe consisting of an electron confined in a one-dimensional infinite square well of width

L = 2.0\;\text{nm}

. The electron serves as a quantum sensor by measuring voltage differences through the Stark effect - when an external electric field

\mathcal{E}

is applied across the well, it creates a perturbation to the quantum energy levels. The perturbation Hamiltonian due to the electric field is

\hat{H}' = e\mathcal{E}x

, where

x

is measured from the left wall of the well and

e

is the elementary charge. For small electric fields, the first-order energy correction to the ground state is given by:

\Delta E_1^{(1)} = \langle \psi_1^{(0)} | \hat{H}' | \psi_1^{(0)} \rangle

where

\psi_1^{(0)}(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{\pi x}{L}\right)

is the unperturbed ground state wavefunction. (a) Calculate the first-order energy shift

\Delta E_1^{(1)}

in terms of

e

\mathcal{E}

, and

L

. (b) The oscilloscope can detect energy shifts as small as

\Delta E_{\text{min}} = 1.0 \times 10^{-22}\;\text{J}

. What is the minimum electric field strength

\mathcal{E}_{\text{min}}

that this quantum probe can detect? Given:

e = 1.6 \times 10^{-19}\;\text{C}

L = 2.0\;\text{nm}