Chapter 39: Q4P (page 1215)

An electron, trapped in a one-dimensional infinite potential well 250 pm wide, is in its ground state. How much energy must it absorb if it is to jump up to the state with $n = 4$ ?

Short Answer

Expert verified

90.3 eV

Step by step solution

Given data

The width of the potential well is

$L = 250 pm = 0.250 nm n = 4$

Energy in a potential well

The $n^{t h}$ state energy of a particle of mass in an infinite potential well of width L is

$E_{n} = (\frac{n^{2} h^{2}}{8 m L}) . . . . . . . . . . (1)$

Determining the energy required to jump from ground state to the fourth state

Here h is the Planck's constant having value

$h = 6.626 \times 10^{- 34} J . s and c = 2.998 \times 10^{8} m / s h_{c} = \frac{(6.626 \times 10^{- 34} J . s) (2.998 \times 10^{8} m / s)}{(1.602 \times 10^{- 19} J / eV) (10^{- 9} m / nm)} = 1240 eV . nm$

Using the value for an electron from Table 37-3 $(511 \times 10^{3} eV)$ , Eq.39 – 4 can be rewritten as

$E_{n} = \frac{n^{2} h^{2}}{8 m L^{2}} = \frac{n^{2} {(h_{c})}^{2}}{8 (m c^{2}) L^{2}}$

The absorbed energy is

$∆ E = E_{4} - E_{1} = \frac{(4^{2} - 1^{2}) {(h_{c})}^{2}}{8 (m_{e} c^{2}) L^{2}} = \frac{15 {(1240 eV . nm)}^{2}}{8 (511 \times 10^{3} eV) {(0.250 nm)}^{2}} = 90.3 eV$

Hence, the required energy is $90.3 eV$

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An electron, trapped in a one-dimensional infinite potential well 250 pm wide, is in its ground state. How much energy must it absorb if it is to jump up to the state with $n = 4$ ?

Short Answer

Step by step solution

Given data

Energy in a potential well

Determining the energy required to jump from ground state to the fourth state

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An electron, trapped in a one-dimensional infinite potential well 250 pm wide, is in its ground state. How much energy must it absorb if it is to jump up to the state with n=4?

Short Answer

Step by step solution

Given data

Energy in a potential well

Determining the energy required to jump from ground state to the fourth state

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Mechanics and Materials

Waves Physics

Radiation

Wave Optics

Particle Model of Matter

Electricity

Study anywhere. Anytime. Across all devices.

An electron, trapped in a one-dimensional infinite potential well 250 pm wide, is in its ground state. How much energy must it absorb if it is to jump up to the state with $n = 4$ ?