Question

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

Short answer

a) 1 b) 2 c) 4 d) 8 Answer: c) \(n=4\)

Step-by-step solution

We know the wavelength of the electron is \(\alpha / 2\). We also know the relationship between the wavelength of an electron and its quantum state in an infinite potential well: \(\lambda_n = \dfrac{2 \alpha}{n}\) Now, we will substitute the given wavelength into the formula and solve for n. #Step 2: Substitute the given wavelength into the formula and solve for n#

We know that the wavelength of the electron is \(\alpha / 2\). Therefore, we can write: \(\dfrac{\alpha}{2} = \dfrac{2 \alpha}{n}\) Now, we will solve for n by cross-multiplying and simplifying: \(n (\dfrac{\alpha}{2}) = 2\alpha\) \(n \alpha = 4\alpha\) Now, we can divide both sides by \(\alpha\) to solve for n: \(n = 4\) So, the electron is in the state of n=4. The correct answer is (c) \(n=4\).