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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

Consider an electron that is confined to the \(x y\) -plane by a two-dimensional rectangular infinite potential well. The width of the well is \(w\) in the \(x\) -direction and \(2 w\) in the \(y\) direction. What is the lowest energy that is shared by more than one distinct state, that is, where two different states have the same energy?

A particle of energy \(E=5 \mathrm{eV}\) approaches an energy barrier of height \(U=8 \mathrm{eV}\). Quantum mechanically there is a finite probability that the particle tunnels through the barrier. If the barrier height is slowly decreased, the probability that the particle will reflect from the barrier will a) decrease. b) increase. c) not change.

State whether each of the following statements is true or false a) In a one-dimensional quantum harmonic oscillator, the energy levels are evenly spaced. b) In an infinite one-dimensional potential well, the energy levels are evenly spaced. c) The minimum total energy possible for a classical harmonic oscillator is zero. d) The correspondence principle states that because the minimum possible total energy for the classical simple harmonic oscillator is zero, the expected value for the fundamental state \((n=0)\) of the one-dimensional quantum harmonic oscillator should also be zero. e) The \(n=0\) state of the one-dimensional quantum harmonic oscillator is the state with the minimum possible uncertainty \(\Delta x \Delta p\)

An oxygen molecule has a vibrational mode that behaves approximately like a simple harmonic oscillator with frequency \(2.99 \cdot 10^{14} \mathrm{rad} / \mathrm{s} .\) Calculate the energy of the ground state and the first two excited states.

Is it possible for the expectation value of the position of an electron to occur at a position where the electron's probability function, \(\Pi(x)\), is zero? If it is possible, give a specific example.
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