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Chapter 39: Q6P (page 1215)

A proton is confined to a one-dimensional infinite potential well 100pm wide. What is its ground-state energy?

Short Answer

Expert verified

0.0205eV

Step by step solution

01

Given data

The width of the potential well is

$L = 100 p m = 100 \times (1 p m \times \frac{1 m}{10^{12} p m}) = 100 \times 10^{- 12} m$

02

Energy in a potential well

The ground state energy of a particle of mass m in an infinite potential well of width L is

$E_{0} = (\frac{h^{2}}{8 m_{e} L^{2}}) n^{2} . . . . . . . . . (1)$

Here h is the Planck's constant having value

$h = 6.63 \times 10^{- 34} J . s$

03

Determining the ground state energy of proton

The mass of the proton is

$m_{p} = 1.67 \times 10^{- 27} k g$

From equation (I) the ground state energy of proton is

$E_{1} = \frac{(6.63 \times 10^{- 34} J . s)}{8 \times 1.67 \times 10^{- 27} k g \times (100 \times 10^{- 12} m)} {(1)}^{2} = 3.29 \times 10^{- 21} \times (1 J) . (1 J \times \frac{1 k g . m^{2} / s^{2}}{1 J}) . (1 s^{2}) . (\frac{1}{1 k g}) . (\frac{1}{m^{2}}) = 3.29 \times 10^{- 21} J$

The energy in electron volt is

$E_{1} = 3.29 \times 10^{- 21} J \times (1 J \times \frac{0.624 \times 10^{19} e V}{1 J}) = 0.0205 e V$

The required energy is 0.0205 eV .

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Most popular questions from this chapter

For the hydrogen atom in its ground state, calculate (a) the probability density $ψ^{2} (r)$ and (b) the radial probability density P(r) for r = a, where a is the Bohr radius.

(a) Show that for the region x>L in the finite potential well of Fig. 39-7, $ψ (x) = {De}^{2 kx}$ is a solution of Schrödinger’s equation in its one-dimensional form, where D is a constant and k is positive. (b) On what basis do we find this mathematically acceptable solution to be physically unacceptable?

An electron is contained in the rectangular corral of Fig. 39-13, with widths $L_{x} = 800 pm$ and $L_{y} = 1600 pm$ . What is the electron’s ground-state energy?

Calculate the radial probability density P(r) for the hydrogen atom in its ground state at (a) r = 0 , (b) r = a , and (c) r = 2a, where a is the Bohr radius.

A proton and an electron are trapped in identical one-dimensional infinite potential wells; each particle is in its ground state. At the center of the wells, is the probability density for the proton greater than, less than, or equal to that of the electron?

See all solutions

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