Chapter 40: Q. 19 (page 1175)

An electron confined in a harmonic potential well emits a 1200nm photon as it undergoes a $3 \to 2$ quantum jump. What is the spring constant of the potential well?

Short Answer

Expert verified

Spring constant of potential well = $2.243 N / m$

Step by step solution

Step 1. Given information

Energy levels of quantum harmonic oscillator,

$E_{n} = (n - \frac{1}{2}) ℏ ω$

Here,

$n$ denotes the state,

$ℏ =$ reduced Planck's constant and

$ω =$ classical angular frequency.

Step 2. Energy of second level and third level

for second level,

$E_{2} = (2 - \frac{1}{2}) ℏ ω$

$= \frac{3}{2} ℏ ω$

for third level,

$E_{3} = (3 - \frac{1}{2}) ℏ ω$

$= \frac{5}{2} ℏ ω$

$Δ E_{3 \to 2} = E_{3} - E_{2}$

$= \frac{5}{2} ℏ ω - \frac{3}{2} ℏ ω$

$= ℏ ω$

Step 3. For classical angular frequency

$E_{photon} = \frac{h c}{λ_{photon}}$

$Δ E_{3 \to 2} = \frac{h c}{λ_{photon}}$

therefore,

$h ω = \frac{h c}{λ_{photon}}$

$\frac{h ω}{2 π} = \frac{h c}{λ_{photon}}$

$ω = \frac{2 π c}{λ_{photon}}$

$ω = \frac{(2 π) (3 \times 10^{8} m / s)}{1200 nm (\frac{10^{- 9} m}{1.0 nm})}$

$= \frac{18.84 \times 10^{8} m / s}{1.2 \times 10^{- 6} m}$

$= 15.7 \times 10^{14} s^{- 1}$

Step 4. Finding spring constant

$ω = \sqrt{\frac{k}{m}}$

$\frac{k}{m} = ω^{2}$

$k = m ω^{2}$

$k = (9.1 \times 10^{- 31} kg) {(2.5 \times 10^{14} s^{- 1})}^{2}$

$= 2.243 N / m$

Thus, the spring constant of potential well $= 2.243 N / m$

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An electron confined in a harmonic potential well emits a 1200nm photon as it undergoes a $3 \to 2$ quantum jump. What is the spring constant of the potential well?

Short Answer

Step by step solution

Step 1. Given information

Step 2. Energy of second level and third level

Step 3. For classical angular frequency

Step 4. Finding spring constant

One App. One Place for Learning.

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An electron confined in a harmonic potential well emits a 1200nm photon as it undergoes a 3→2quantum jump. What is the spring constant of the potential well?

Short Answer

Step by step solution

Step 1. Given information

Step 2. Energy of second level and third level

Step 3. For classical angular frequency

Step 4. Finding spring constant

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Quantum Physics

Linear Momentum

Work Energy and Power

Particle Model of Matter

Force

Nuclear Physics

Study anywhere. Anytime. Across all devices.

An electron confined in a harmonic potential well emits a 1200nm photon as it undergoes a $3 \to 2$ quantum jump. What is the spring constant of the potential well?