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Chapter 2: Problem 10

A particle of mass \(m\) is confined to a one-dimensional box of length \(L\). Calculate the probability of finding it in the following regions: (a) \(0 \leq x \leq \frac{1}{2} L,\) (b) \(0 \leq x \leq \frac{1}{4} L\) (c) \(\frac{1}{2} L-\delta x \leq x \leq \frac{1}{2} L+\delta x .\) Derive expressions for a general value of \(n\). Then evaluate the probabilities (i) for \(n=1\) (ii) in the limit \(n \rightarrow \infty\). Compare the latter to the classical expectations.

Short Answer

Expert verified
The probability of finding the particle will depend on the given intervals and the value of \(n\). For very big values of \(n\) the result will tend closer and closer to the classical expectation of equal probability throughout the box.

Step by step solution

01

Calculate the Probability for Part (a)

The probability \(P\) for finding the particle in region (a) can be evaluated with the following integral: \(P = \int_0^{L/2} |\Psi |^2 dx = \int_0^{L/2} \frac{2}{L} \sin^2(\frac{n\pi x}{L}) dx\). Solve the integral to get the probability.
02

Calculate the Probability for Part (b)

The probability \(P\) for finding the particle in region (b) can be evaluated with the following integral: \(P = \int_0^{L/4} |\Psi |^2 dx = \int_0^{L/4} \frac{2}{L} \sin^2(\frac{n\pi x}{L}) dx\). Again, solve the integral to get the probability.
03

Calculate the Probability for Part (c)

The probability \(P\) for finding the particle in region (c) can be evaluated with the following integral: \(P = \int_{L/2-\delta x}^{L/2+\delta x} |\Psi |^2 dx = \int_{L/2-\delta x}^{L/2+\delta x} \frac{2}{L} \sin^2(\frac{n\pi x}{L}) dx\). Again, solve the integral to get the probability.
04

Evaluate for \(n=1\)

Substitute \(n=1\) into the general expression for the probability in step 1, step 2 and step 3 to evaluate the probabilities.
05

Evaluate in the Limit \(n \rightarrow \infty\)

Evaluate the probabilities in the limit as \(n\) approaches infinity and compare the results with classical expectations, in which the particle is equally likely to be in any part of the box.

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Most popular questions from this chapter

Calculate the values of (a) \(\langle x\rangle\) (b) \(\left\langle x^{2}\right\rangle\) \(,(\mathrm{c})\left\langle p_{x}\right\rangle,(\mathrm{d})\left\langle p_{x}^{2}\right\rangle\) for a harmonic oscillator in its ground state by evaluation of the appropriate integrals (as in Problems \(2.13-2.15\) ). Examine the value of \(\Delta x \Delta p_{x}\) in the light of the uncertainty principle. Hint. Use the integrals \\[ \begin{array}{l} \int_{-\infty}^{\infty} \mathrm{e}^{-\alpha x^{2}} \mathrm{d} x=\left(\frac{\pi}{\alpha}\right)^{1 / 2} \\ \int_{0}^{\infty} x \mathrm{e}^{-\alpha x^{2}} \mathrm{d} x=\frac{1}{2 \alpha} \\\ \int_{-\infty}^{\infty} x^{2} \mathrm{e}^{-\alpha x^{2}} \mathrm{d} x=\frac{1}{2}\left(\frac{\pi}{\alpha^{3}}\right)^{1 / 2} \end{array} \\]

Calculate the energies and wavefunctions for a particle in a one-dimensional square well in which the potential energy rises to a finite value \(V\) at each end, and is zero inside the well; that is \\[ \begin{array}{ll} V(x)=V & x \leq 0 \text { and } x \geq L \\ V(x)=0 & 0

An electron is confined to a one-dimensional box of length \(L .\) What should be the length of the box in order for its zero-point energy to be equal to its rest mass energy \(\left(m_{\mathrm{e}} c^{2}\right) ?\) Express the result in terms of the Compton wavelength, \(\lambda_{\mathrm{C}}=h / m_{\mathrm{e}} c\)

A particle was prepared travelling to the right with all momenta between \(\left(k-\frac{1}{2} \Delta k\right) \hbar\) and \(\left(k+\frac{1}{2} \Delta k\right) \hbar\) contributing equally to the wavepacket. Find the explicit form of the wavepacket at \(t=0,\) normalize it, and estimate the range of positions, \(\Delta x,\) within which the particle is likely to be found. Compare the last conclusion with a prediction based on the uncertainty principle. Hint. Use eqn 2.13 with \(g=B\) a constant, inside the range \(k-\frac{1}{2} \Delta k\) to \(k+\frac{1}{2} \Delta k\) and zero elsewhere, and eqn 2.12 with \(t=0\) for \(\Psi_{k} .\) To evaluate \(\int\left|\Psi_{k}\right|^{2} \mathrm{d} \tau\) (for the normalization step) use the integral \(\int_{-\infty}^{\infty}(\sin x / x)^{2} \mathrm{d} x=\pi .\) Take \(\Delta x\) to be determined (numerically) by the locations where \(|\Psi|^{2}\) falls to half its value at \(x=0\) For the last part use \(\Delta p_{x} \approx \hbar \Delta k\)

Consider a harmonic oscillator of mass \(m\) undergoing harmonic motion in two dimensions \(x\) and \(y .\) The potential energy is given by \(V(x, y)=\frac{1}{2} k_{x} x^{2}+\frac{1}{2} k_{y} y^{2}\). (a) Write down the expression for the hamiltonian operator for such a system. (b) What is the general expression for the allowable energy levels of the two- dimensional harmonic oscillator? (c) What is the energy of the ground state (the lowest energy state)? Hint. The hamiltonian operator can be written as a sum of operators.
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