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Chapter 3: Problem 8

Evaluate the probability that a particle of mass \(m\) in the ground state of a circular square well of radius \(a\) will be found within the circular area of radius \(1 / 2 a\)

Short Answer

Expert verified
The required probability that the particle will be found within the circular area of radius \(1 / 2 a\) can be found by integrating the probability distribution function from \(r=0\) to \(r=1 / 2 a\) over all angles from \(0\) to \(2\pi\).

Step by step solution

01

Getting the ground state wave function

The wave function for a particle in a circular square well is given by the Bessel function, which for the ground state (n=1) is a circularly symmetric wave, represented as: \(\Psi(r)=A J_0(kr)\), where \(A\) is normaliztion constant, \(J_0\) is the Bessel function of the first kind and \(kr=ka\) is the root of the Bessel function that is confined within the well. Here, we need to determine \(A\) such that the total probability over the well is unity: \(\int_0^{2\pi} \int_0^a |\Psi|^2 r dr d\phi=1\). This gives us \(A= \sqrt{2/a}\), so \(\Psi(r)=\sqrt{2/a} J_0(kr)\).
02

Calculate the Probability Distribution function

The probability distribution function can be obtained by squaring the absolute value of the wave function. Hence, the probability distribution function is \(P(r)=|\Psi(r)|^2 = (2/a) J_0^2(kr)\).
03

Evaluate the Probability that the particle lies within \(1 / 2 a\)

We can determine the probability that the particle is within a circular area of radius \(1 / 2 a\) by integrating the probability distribution from \(r=0\) to \(r=1 / 2 a\) over all angles from \(0\) to \(2\pi\). Hence, the required probability is given by \(P=\int_0^{2\pi} \int_0^{1 / 2 a} (2/a) J_0^2(kr) r dr d\phi\).

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Most popular questions from this chapter

(a) Confirm that the radius of gyration of a solid uniform sphere of radius \(R\) is \(r=\left(\frac{2}{5}\right)^{1 / 2} R .\) (b) What is the radius of gyration of a solid uniform cylinder of radius \(R\) and length \(l\) about an axis perpendicular to its principal axis? Hint. For part (a), the moment of incrtia of a sphere is \(^{2} / s M R^{2}\)

(a) Construct a wavepacket \(\Psi=\mathrm{N} \sum_{m_{\mathrm{r}}=0}^{\infty}\left(1 / m_{l} !\right) \mathrm{e}^{\mathrm{in} \varphi}\) and normalize it to unity, Sketch the form of \(|\Psi|^{2}\) for \(0 \leq \varphi \leq 2 \pi\) (b) Calculate \(\langle\varphi\rangle,\langle\sin \varphi\rangle,\) and \(\left\langle l_{z}\right\rangle\) (c) Why is \(\left\langle l_{z}\right\rangle \leq \hbar ?\) Hint. Draw on a variety of pieces of information, including \(\sum_{n=0}^{\infty} x^{n} / n !=\mathrm{e}^{x}\) and the following integrals: $$\int_{0}^{2 \pi} \mathrm{e}^{\operatorname{scos} \phi} \mathrm{d} \varphi=2 \pi I_{0}(z) \int_{0}^{2 \pi} \cos \varphi \mathrm{e}^{\sec \varphi} \mathrm{d} \varphi=2 \pi I_{1}(z)$$ with \(I_{0}(2)=2.280 \ldots, I_{1}(2)=1.591 \ldots ;\) the \(I(z)\) are modificd Bessel functions.

Calculate (a) the mean radius, (b) the mean square radius, and (c) the most probable radius of the 1 s- \(, 2\) sand 3 s-orbitals of a hydrogenic atom of atomic number \(Z\) Hint. For the most probable radius look for the principal maximum of the radial distribution function. You will find the following integral useful: $$\int_{0}^{\infty} x^{n} \mathrm{c}^{-\alpha x} \mathrm{d} x=\frac{n !}{a^{n+1}}$$

Calculate the angle that the angular momentum vector makes with the \(z\) -axis when the system is described by the wavefunction \(\psi_{l m_{i}}\). Show that the minimum angle approaches zero as \(l\) approaches infinity. Calculate the allowed angles when \(l\) is \(1,2,\) and 3
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