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Chapter 37: Problem 1

The wavelength of an electron in an infinite potential is \(\alpha / 2,\) where \(\alpha\) is the width of the infinite potential well. Which state is the electron in? a) \(n=3\) b) \(n=6\) c) \(n=4\) d) \(n=2\)

Short Answer

Expert verified
Answer: c) \(n=4\)

Step by step solution

01

Write down the formula for the wavelength of an electron in an infinite potential well.

The wavelength of an electron in an infinite potential well can be expressed as \(\lambda_n = \frac{2\alpha}{n}\), where \(\lambda_n\) denotes the wavelength of an electron in the nth state, \(\alpha\) is the width of the potential well, and n is an integer representing the quantum state.
02

Plug in the given wavelength.

We are given that \(\lambda = \frac{\alpha}{2}\). Now we want to find the value of n for which this equation is true.
03

Solve for n.

We want to find n, such that \(\frac{\alpha}{2} = \frac{2\alpha}{n}\). We can rearrange this equation as follows: $$ n = \frac{2\alpha}{\frac{\alpha}{2}} $$
04

Simplify the equation.

Now, we have: $$ n = \frac{2\alpha \times 2}{\alpha} $$
05

Cancel out the common factors.

In the expression above, we can cancel out the alpha terms: $$ n = \frac{2\cancel{\alpha} \times 2}{\cancel{\alpha}}$$
06

Find the value of n.

Thus, we are left with: $$ n = 2\times 2 = 4$$
07

Match the answer with the given options.

We found that the electron is in state n = 4. So, the correct answer is: c) \(n=4\)

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Most popular questions from this chapter

Particle-antiparticle pairs are occasionally created out of empty space. Looking at energy-time uncertainty, how long would such particles be expected to exist if they are: a) an electron/positron pair? b) a proton/antiproton pair?

A particle is in an infinite square well of width \(L\) and is in the \(n=3\) state. What is the probability that, when observed, the particle is found to be in the rightmost \(10.0 \%\) of the well?

Example 37.1 calculates the energy of the wave function with the lowest quantum number for an electron confined to a box of width \(2.00 \AA\) in the one-dimensional case. However, atoms are three-dimensional entities with a typical diameter of \(1.00 \AA=10^{-10} \mathrm{~m} .\) It would seem then that the next, better approximation would be that of an electron trapped in a three-dimensional infinite potential well (a potential cube with sides of \(1.00 \mathrm{~A}\) ). a) Derive an expression for the electron wave function and the corresponding energies for a particle in a three dimensional rectangular infinite potential well. b) Calculate the lowest energy allowed for the electron in this case.

Simple harmonic oscillation occurs when the potential energy function is equal to \((1 / 2) k x^{2},\) where \(k\) is a constant. What happens to the ground state energy level if \(k\) is increased? a) It increases. b) It remain the same. c) It decreases.

True or False: The larger the amplitude of a Schrödinger wave function, the larger its kinetic energy. Explain your answer.
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