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Chapter 35: Problem 44

What's the probability of finding a particle in the central \(80 \%\) of an infinite square well, assuming it's in the ground state?

Short Answer

Expert verified
The probability of the particle in the infinite square well's ground state being found within the central \(80\%\) of the well can be calculated through these steps. Given the integral calculus and quantum mechanics principles, the final solution would need to be obtained by evaluating the integral in Step 4.

Step by step solution

01

Expression of the wave function

The first step involves understanding the wave function of a particle in the ground state of an infinite square well. The function, when the well's width is considered as \(a\) and the particle is in the ground state (n=1), can be expressed as: \[ \psi_1(x) = \sqrt{\dfrac{2}{a}} \sin\left(\dfrac{\pi x}{a}\right) \] This wave function is zero at the boundaries of the well (x=0 and x=a).
02

Setting up the probability density function

The probability density function, which represents the probability of finding a particle in a certain position, is calculated by squaring the magnitude of the wave function (|蠄(x)|^2). As such, it can be expressed as: \[ P(x) = |\psi(x)|^2 = \dfrac{2}{a} \sin^2\left(\dfrac{\pi x}{a}\right) \] The normalization concept here guarantees that the integral of this function across all space (or within the confines of this well) equals 1.
03

Setting up the limits for integration

Since the problem is asking for the probability of the particle being found within the central \(80 \%\) of the well, we need to set up our integration limits accordingly. The integration should be performed from \(0.1a\) to \(0.9a\), which represents the central \(80 \%\) of the width of the well.
04

Performing the integration

Perform the integration of the probability density function with the set limits, like so: \[ P = \int_{0.1a}^{0.9a} \dfrac{2}{a} \sin^2\left(\dfrac{\pi x}{a}\right) dx \] Calculate the above integral to find the desired probability, applying basic integral calculus principles and substitution methodology.
05

Calculating and interpreting the result

Calculate the result from Step 4. The computed outcome will represent the probability of the particle being found within the central \(80 \%\) of the infinite square well when it is in its ground state. The probability should be interpreted in accordance with the principles of quantum mechanics 鈥� A higher value suggests a higher likelihood of finding the particle within this prescribed interval.

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Most popular questions from this chapter

A particle's wave function is \(\psi=A e^{-x^{2} / a^{2}},\) where \(A\) and \(a\) are constants. (a) Where is the particle most likely to be found? (b) Where is the probability per unit length half its maximum value?

In terms of de Broglie's matter-wave hypothesis, how does making the sides of a box different lengths remove the degeneracy associated with a particle confined to that box?

Show explicitly that the difference between adjacent energy levels in an infinite square well becomes arbitrarily small compared with the energy of the upper level, in the limit of large quantum number \(n\)

(a) Using the potential energy \(U=\frac{1}{2} m \omega^{2} x^{2}\) discussed on page \(675,\) develop the Schr枚dinger equation for the harmonic oscillator. (b) Show by substitution that \(\psi_{0}(x)=A_{0} e^{-\alpha^{2} x^{2} / 2}\) satisfies your equation, where \(\alpha^{2}=m \omega / \hbar\) and the energy is given by Equation 35.7 with \(n=0 .\) (c) Find the normalization constant \(A_{0}\) You then have the ground-state wave function for the harmonic oscillator.

An infinite square well extends from \(-L / 2\) to \(L / 2 .\) (a) Find expressions for the normalized wave functions for a particle of mass \(m\) in this well, giving separate expressions for even and odd quantum numbers. (b) Find the corresponding energy levels.
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