The Quantum Pressure Sensor · AI Generator #GEN-22D54332

Physics · quantum-mechanics

Problem

A novel quantum pressure sensor consists of an electron trapped in a one-dimensional infinite square well of width

L = 2.0\;\text{nm}

. The well is created between two conducting plates, and when external pressure is applied, the well width decreases according to

L(P) = L_0(1 - \alpha P)

, where

\alpha = 1.2 \times 10^{-10}\;\text{Pa}^{-1}

and

L_0 = 2.0\;\text{nm}

is the initial width at zero pressure. The sensor works by measuring the photon energy emitted when the electron transitions from the

n=2

state to the

n=1

state. At zero pressure, this transition emits a photon of energy

E_0 = 0.565\;\text{eV}

. (a) What is the pressure

P

when the emitted photon energy increases to

E = 0.600\;\text{eV}

? (b) What is the fractional change in the quantum mechanical force exerted by the electron on the walls when the pressure changes from

0

to the value found in part (a)? The quantum force on each wall is

F = -\frac{\partial E_n}{\partial L}

where

E_n

is the energy of the occupied state. Given:

\hbar = 1.055 \times 10^{-34}\;\text{J·s}

m_e = 9.11 \times 10^{-31}\;\text{kg}

1\;\text{eV} = 1.602 \times 10^{-19}\;\text{J}