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Chapter 37: Problem 48

Find the probability of finding an electron trapped in a one-dimensional infinite well of width \(2.00 \mathrm{nm}\) in the \(n=2\) state between 0.800 and \(0.900 \mathrm{nm}\) (assume that the left edge of the well is at \(x=0\) and the right edge is at \(x=2.00 \mathrm{nm}\) ).

Short Answer

Expert verified
Answer: The probability of finding an electron in the n=2 state between 0.800 nm and 0.900 nm is approximately 0.049874.

Step by step solution

01

Write down the wave function for an electron in an infinite well

The wave function for an electron trapped in a one-dimensional infinite well of width L in the nth state is given by: $$\psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)$$ Here, L=2.00 nm is the width of the well, and n=2 is the energy state.
02

Square the wave function to get the probability density

To calculate the probability of finding an electron in a certain region, we need to find the probability density (the square of the wave function): $$\rho_n(x) = |\psi_n(x)|^2 = \frac{2}{L}\sin^2\left(\frac{n\pi x}{L}\right)$$
03

Write the probability density function for n=2 and L=2.00 nm

Substitute n=2 and L=2.00 nm into the probability density function: $$\rho_2(x) = \frac{2}{2.00 \mathrm{nm}}\sin^2\left(\frac{2\pi x}{2.00 \mathrm{nm}}\right)$$
04

Integrate the probability density function over the specified range

To find the probability of finding the electron between 0.800 nm and 0.900 nm, integrate the probability density function over this range: $$P = \int_{0.800 \mathrm{nm}}^{0.900 \mathrm{nm}} \rho_2(x) dx$$
05

Simplify and evaluate the integral

Use the antiderivative identity: $$\int \sin^2(ax) dx = \frac{1}{4a}\left(x - \frac{1}{2a}\sin(2ax)\right) + C$$ Substitute a=2π/2 and the limits of integration: $$P = \left[\frac{1}{4(2\pi/2)}\left(x - \frac{1}{2(2\pi/2)}\sin\left(2\pi x\right)\right)\right]_{0.800 \mathrm{nm}}^{0.900 \mathrm{nm}}$$ Evaluate the expression: $$P \approx 0.049874$$ The probability of finding an electron in the n=2 state between 0.800 nm and 0.900 nm is approximately 0.049874.

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Most popular questions from this chapter

An electron is in an infinite square well of width \(a\) : \((U(x)=\infty\) for \(x<0\) and \(x>a\) ). If the electron is in the first excited state, \(\Psi(x)=A \sin (2 \pi x / a),\) at what position is the probability function a maximum? a) 0 b) \(a / 4\) c) \(a / 2\) d) \(3 a / 4\) and \(3 a / 4\) e) at both \(a / 4\)

A beam of electrons moving in the positive \(x\) -direction encounters a potential barrier that is \(2.51 \mathrm{eV}\) high and \(1.00 \mathrm{nm}\) wide. Each electron has a kinetic energy of \(2.50 \mathrm{eV},\) and the electrons arrive at the barrier at a rate of 1000 electrons/s (1000. electrons every second). What is the rate \(\mathrm{I}_{\mathrm{T}}\) in electrons/s at which electrons pass through the barrier, on average? What is the rate \(\mathrm{I}_{\mathrm{R}}\) in electrons/s at which electrons reflect back from the barrier, on average? Determine and compare the wavelengths of the electrons before and after they pass through the barrier.

Find the ground state energy (in units of eV) of an electron in a one- dimensional quantum box, if the box is of length \(L=0.100 \mathrm{nm}\).

An electron is confined between \(x=0\) and \(x=L\). The wave function of the electron is \(\psi(x)=A \sin (2 \pi x / L)\). The wave function is zero for the regions \(x<0\) and \(x>L\) a) Determine the normalization constant \(A\). b) What is the probability of finding the electron in the region \(0 \leq x \leq L / 3 ?\)

A surface is examined using a scanning tunneling microscope (STM). For the range of the working gap, \(L\), between the tip and the sample surface, assume that the electron wave function for the atoms under investigation falls off exponentially as \(|\Psi|=e^{-\left(10.0 \mathrm{nm}^{-1}\right) a}\). The tunneling current through the STM tip is proportional to the tunneling probability. In this situation, what is the ratio of the current when the STM tip is \(0.400 \mathrm{nm}\) above a surface feature to the current when the tip is \(0.420 \mathrm{nm}\) above the surface?
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