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Chapter 35: Problem 40

Sketch the probability density for the \(n=2\) state of an infinite square well extending from \(x=0\) to \(x=L\), and determine where the particle is most likely to be found.

Short Answer

Expert verified
The particle in the \(n=2\) state of an infinite square well extending from \(x=0\) to \(x=L\) is most likely to be found at positions \(L/3\) and \(2L/3\). The probability density function of the \(n=2\) state can be represented as \(P_2(x) = \frac{2}{L} \sin^2\left(\frac{2\pi x}{L}\right)\).

Step by step solution

01

Identification of energy eigenfunction for the infinite square well

The infinite square well is a potential energy profile where the energy is infinite at the boundaries, confining the particle to move between \(x=0\) and \(x=L\) only. The energy eigenfunction for a particle in an infinite square well is given by: \[\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)\]where \(L\) is the width of the well and \(n\) denotes the state of the particle. This eigenfunction represents the state of the particle with respect to location \(x\). For the exercise in question, \(n=2\).
02

Calculate the Probability Density Function (PDF)

In quantum mechanics, to get the probability density function, we square the wave function (recall that the wave function is complex, and the modulus squared of a complex function gives us a real probability density). The probability density function for this particular problem is: \[P_n(x) = |\psi_n(x)|^2 = \left(\sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)\right)^2 = \frac{2}{L} \sin^2\left(\frac{n\pi x}{L}\right)\]Substituting \(n=2\), we get the probability density function for the \(n=2\) state as:\[P_2(x) = \frac{2}{L} \sin^2\left(\frac{2\pi x}{L}\right)\]
03

Determine the position with the highest probability

The position at which the particle is most likely to be found corresponds to the maximum of the PDF. However, for \(n=2\), the PDF has two equal maximum points, \(L/3\) and \(2L/3\). Hence, the particle in the \(n=2\) state is most likely to be found in either of these positions.

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Most popular questions from this chapter

The solution to the Schr枚dinger equation for a particular potential is \(\psi=0\) for \(|x|>a\) and \(\psi=A \sin (\pi x / a)\) for \(-a \leq x \leq a\) where \(A\) and \(a\) are constants. In terms of \(a,\) what value of \(A\) is required to normalize \(\psi\) ?

Is quantization significant for macromolecules confined to biological cells? To find out, consider a protein of mass \(250,000 \mathrm{u}\) confined to a 10 - \(\mu\) m-diameter cell. Treating this as a particle in a one-dimensional square well, find the energy difference between the ground state and the first excited state. Given that biochemical reactions typically involve energies on the order of \(1 \mathrm{eV},\) what do you conclude about the role of quantization?

A particle is confined to a two-dimensional box whose sides are in the ratio \(1: 2 .\) Are any of its energy levels degenerate? If so, give an example. If not, why not?

Does quantum tunneling violate energy conservation? Explain.

An electron is in a narrow molecule \(4.4 \mathrm{nm}\) long, a situation that approximates a one-dimensional infinite square well. If the electron is in its ground state, what is the maximum wavelength of electromagnetic radiation that can cause a transition to an excited state?
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