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Chapter 1: Problem 6

A particle in an infinite one-dimensional system was described by the wavefunction \(\psi(x)=\mathrm{Ne}^{-x^{2} / 2 r^{2}}\). Normalize this function. Calculate the probability of finding the particle in the range \(-\Gamma \leq x \leq \Gamma\). Hint. The integral encountered in the second part is the error function. It is available in mathematical software.

Short Answer

Expert verified
The normalized function is \( \psi_{normalized}(x) = e^{-x^{2} / 2 r^{2}}/\sqrt{π}r \) and probability of finding the particle in the range \( -\Gamma \leq x \leq \Gamma \) is \( \frac{2}{\sqrt{π}} erf(\Gamma / r) \).

Step by step solution

01

Normalizing the function

To normalize the function, firstly need to find \( |\psi(x)|^{2} \) which is \( |Ne^{-x^{2} / 2 r^{2}}|^{2} = N^{2}e^{-x^{2}/r^{2}} \). Then, we set the integral over all space of this equal to one, \[ \int_{-\infty}^{\infty} N^{2}e^{-x^{2}/r^{2}} dx = 1 \]. Solving the integral gives \( N^{2}\sqrt{π}r = 1 \). Hence, \( N = 1/\sqrt{π}r \) and our normalized wave function is \( \psi_{normalized}(x) = e^{-x^{2} / 2 r^{2}}/\sqrt{π}r \).
02

Computing the probability

Now that we have the normalized wave function, we can compute the probability of finding the particle in the range \( -\Gamma \leq x \leq \Gamma \) by integrating the probability density over that range. The probability is computed as \[ \int_{-\Gamma}^{\Gamma} |\psi_{normalized}(x)|^{2} dx = \int_{-\Gamma}^{\Gamma} \frac{e^{-x^{2}/r^{2}}}{πr^{2}} dx \]. The integral here is the error function, often denoted as \( erf(x) \), and its value can be found using mathematical software. Hence the result will be \( \frac{2}{\sqrt{π}} erf(\Gamma / r) \).

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Most popular questions from this chapter

Construct quantum mechanical operators in the position representation for the following observables: (a) kinetic energy in one and in three dimensions, (b) the inverse separation, \(1 / x,\) (c) electric dipole moment \(\left(\Sigma_{i} Q_{i} r_{i} \text { where } r_{i}\right.\) is the position of a charge \(Q_{i}\) ), (d) \(z\) -component of angular momentum \(\left(x p_{y}-y p_{x}\right),\) (e) the mean square deviations of the position and momentum of a particle from the mean values.

Evaluate the commutator \(\left[l_{y}\left[l_{y}, l_{z}\right]\right]\) given that \(\left[l_{x}, l_{y}\right]=i \hbar l,\left[l_{y}, l_{z}\right]=i \hbar l_{x},\) and \(\left[l_{z}, l_{x}\right]=i \hbar l_{y}\)

Show that (a) \([A, B]=-[B, A],\) (b) \(\left[A^{m}, A^{n}\right]=0\) for all \(m, n,\) (c) \(\left[A^{2}, B\right]=A[A, B]+[A, B] A\) (d) \([A,[B, C]]+[B,[C, A]]+[C,[A, B]]=0\)

Evaluate the commutators (a) \(\left[(1 / x), p_{x}\right]\) (b) \(\left[(1 / x), p_{x}^{2}\right]\) (c) \(\left[x p_{y}-y p_{x}, y p_{z}-z p_{y}\right]\) (d) \(\left[x^{2}\left(\partial^{2} / \partial y^{2}\right), y(\partial / \partial x)\right]\)

Evaluate the expectation values of the operators \(p_{x}\) and \(p_{x}^{2}\) for a particle with wavefunction \((2 / L)^{1 / 2} \sin (\pi x / L)\) in the range 0 to \(L\)
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