Chapter 6: Q6E (page 1402)

What is the energy difference between the two lowest energy levels for a proton in a cubical box with side length $1.00 x 10 -^{14} m$ , the approximate diameter of a nucleus?

Short Answer

Expert verified

The energy difference between the two lowest energy levels for a proton is $∆ E = 9.89 \times 10^{- 13} J = 6.162 M e V$

Step by step solution

Energy Formula

In a cubical box with side length L, the allowed energies for a particle of mass m are given by;

${E_{n}}_{x}, n_{y}, n_{z} = \frac{(n_{X}^{2} + n_{Y}^{2} + n_{Z}^{2}) π^{2} h^{2}}{2 m L^{2}}$

The energy difference between the two lowest energy levels for a proton

$(n_{X}, n_{Y}, n_{Z}) = (1, 1, 1)$ corresponds to the lowest level, so

$(n_{X}^{2} + n_{Y}^{2} + n_{z}^{2})_{L} = {({(1)}^{2} + (])}^{2} + {(1)}^{2} = 3)$

and one of the combinations $(n_{X}, n_{Y}, n_{z}) = (2, 1, 1)$ , $(n_{X}, n_{Y}, n_{z}) = (1, 2, 1)$ , and $(n_{X}, n_{Y}, n_{Z}) = (1, 1, 2)$ corresponds to the second-lowest energy;

${(n_{x}^{2}, n_{y}^{2}, n_{z}^{2})}_{L} = {(2)}^{2} + {(1)}^{2} + {(1)}^{2} = 6$

The energy separation between these two levels can be calculated using the equation:

$\begin{array}{l} ∆ E = E_{u} - E_{L} = \frac{6 π^{2} h}{2 m_{e} L} - & \frac{3 π^{2} h^{2}}{2 m_{e} L^{2}} \\ ∆ E = \frac{3 π^{2} h^{2}}{2 m_{e} L^{2}} \end{array}$

Putting the values together now;

$∆ E = \frac{3 π^{2} {(1.055 \times 10^{- 34} J . s)}^{2}}{2 (1.67 \times 10^{- 27} k g) {(1.0 \times 10^{- 14} m)}^{2}} = 9.89 \times 10^{- 13} ∆ E = 9.89 \times 10^{- 13} J = 6.162 M e V$

Hence, the energy difference between the two lowest energy levels for a proton is $∆ E = 9.89 \times 10^{- 13} J = 6.162 M e V$

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What is the energy difference between the two lowest energy levels for a proton in a cubical box with side length $1.00 x 10 -^{14} m$ , the approximate diameter of a nucleus?

Short Answer

Step by step solution

Energy Formula

The energy difference between the two lowest energy levels for a proton

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What is the energy difference between the two lowest energy levels for a proton in a cubical box with side length1.00x10-14m, the approximate diameter of a nucleus?

Short Answer

Step by step solution

Energy Formula

The energy difference between the two lowest energy levels for a proton

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Wave Optics

Work Energy and Power

Energy Physics

Fields in Physics

Medical Physics

Engineering Physics

Study anywhere. Anytime. Across all devices.

What is the energy difference between the two lowest energy levels for a proton in a cubical box with side length $1.00 x 10 -^{14} m$ , the approximate diameter of a nucleus?