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Chapter 40: Q. 7 (page 1175)

Show that the penetration distance $η$ has units of $m$ .

Short Answer

Expert verified

The penetration distance $η$ has units of $m$ is given below in step.

Step by step solution

01

Given Information

We need to find the penetration distance $η$ has units of $m$ .

02

Explanation

The equation of penetration distance is given as:

$η = \frac{h}{\sqrt{2 m (U_{0} - E)}}$

The unit of role="math" localid="1650172801611" $h$ is $J . s$ . Its dimension can be $[h] = M \times L^{2} \times T^{- 1}$ .

The unit of $m$ is $k g$ , so dimension is $[m] = M$ .

The unit of $(U_{0} - E)$ is $J$ , then dimension is $(U_{0} - E) = M \times L^{2} \times T^{- 2}$

Therefore, the dimension of $η$ is:

$[h] = \frac{M \times L^{2} \times T^{- 1}}{\sqrt{M \times L^{2} \times T^{- 2}}} = \frac{M \times L^{2} \times T^{- 1}}{M \times L \times T^{- 1}} = L$

Then, $η$ has the dimension of length. So its unit is $m$ .

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Most popular questions from this chapter

a. Sketch graphs of the probability density ${|ψ (x)|}^{2}$ for the four states in the finite potential well of Figure $40.14$ a. Stack them vertically, similar to the Figure $40.14$ a graph of $ψ x$ .

b. What is the probability that a particle in the $n = 2$ state of the finite potential well will be found at the center of the well? Explain.

c. Is your answer to part b consistent with what you know about standing waves? Explain.

The energy of an electron in a $2.00 eV$ deep potential well is $1.50 eV$ . At what distance into the classically forbidden region has the amplitude of the wave function decreased to $25 %$ of its value at the edge of the potential well?

Verify that the n=1 wave function $ψ_{1} (x)$ of the quantum harmonic oscillator really is a solution of the Schrödinger equation. That is, show that the right and left sides of the Schrödinger equation are equal if you use the $ψ_{1} (x)$ wave function.

A helium atom is in a finite potential well. The atom’s energy is $1.0 eV$ below $U_{0}$ . What is the atom’s penetration distance into the classically forbidden region?

A particle in a potential well is in the $n = 5$ quantum state. How many peaks are in the probability density $P (x) = {|ψ (x)|}^{2}$ ?

See all solutions

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