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Chapter 37: Problem 16

Sketch the two lowest energy wave functions for an electron in an infinite potential well that is \(20 \mathrm{nm}\) wide and a finite potential well that is \(1 \mathrm{eV}\) deep and is also \(20 \mathrm{nm}\) wide. Using your sketches, can you determine whether the energy levels in the finite potential well will be lower, the same, or higher than in the infinite potential well?

Short Answer

Expert verified
Answer: The energy levels in the finite potential well are lower than in the infinite potential well.

Step by step solution

01

Analyze infinite potential well

An infinite potential well has zero potential energy (V(x)=0) within the well and infinite potential energy outside the well. The width of the well is given as 20 nm. The Schrödinger equation becomes a one-dimensional problem, with the boundary condition of zero probability outside the well. The general solutions of the wave function inside the well are of the form: $$\psi_n(x) = A \sin(k_n x)$$ where \(k_n = \frac{n \pi}{L}\), and \(L\) is the width of the well (20 nm).
02

Sketch lowest energy wave functions for infinite potential well

For an infinite potential well, the two lowest energy wave functions are given by: $$\psi_1(x) = A \sin\left(\frac{\pi x}{L}\right)$$ $$\psi_2(x) = A \sin\left(\frac{2\pi x}{L}\right)$$ We can sketch the wave functions inside the well between 0 and 20 nm. The first wave function, \(\psi_1(x)\), is a single sine wave. The second wave function, \(\psi_2(x)\), is two sine waves within the well.
03

Analyze finite potential well

A finite potential well has zero potential energy within the well (V(x)=0) and a finite potential energy (\(V_0 = 1\) eV) outside the well. The Schrödinger equation becomes again a one-dimensional problem. The width of the well is given as 20 nm. Inside the well, the wave function has the same form as in the infinite potential well. Outside the well, the wave function is a decaying exponential: $$\psi(x) = B e^{-(x-L)/a}$$ where \(a = \frac{\hbar^2}{2m_eV_0}\).
04

Sketch lowest energy wave functions for finite potential well

For a finite potential well, the two lowest energy wave functions are also given by: $$\psi_1(x) = A \sin\left(\frac{\pi x}{L}\right)$$ $$\psi_2(x) = A \sin\left(\frac{2\pi x}{L}\right)$$ Inside the well, the wave functions have the same shape as those in the infinite potential well. However, outside the well, the wave functions are decaying exponentials. We can sketch the wave functions, making sure to include the exponential decay outside the well.
05

Compare energy levels between finite and infinite potential wells

The energy levels for finite and infinite potential wells can be compared by looking at the wave functions. For the infinite potential well, the energy levels are quantized and given by: $$E_n = \frac{n^2\pi^2\hbar^2}{2m_eL^2}$$ For the finite potential well, energy levels are also quantized but are affected by the exponential decay of the wave functions outside the well. The quantization is less strict, and the energy levels will be slightly lower than those in the infinite potential well due to the penetration of the wave function into the barrier. In conclusion, the energy levels in the finite potential well will be lower than in the infinite potential well.

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Most popular questions from this chapter

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 \(x=2.00 \mathrm{nm}\) ).

The ground state wave function for a harmonic oscillator is given by \(\Psi_{0}(x)=A_{2} e^{-x^{2} / 2 b^{2}}\). a) Determine the normalization constant \(A\). b) Determine the probability that a quantum harmonic oscillator in the \(n=0\) state will be found in the classically forbidden region.

Example 37.1 calculates the energy of the wave function with the lowest quantum number for an electron confined to a box of width \(2.00 \AA\) in the one-dimensional case. However, atoms are three-dimensional entities with a typical diameter of \(1.00 \AA=10^{-10} \mathrm{~m} .\) It would seem then that the next, better approximation would be that of an electron trapped in a three-dimensional infinite potential well (a potential cube with sides of \(1.00 \mathrm{~A}\) ). a) Derive an expression for the electron wave function and the corresponding energies for a particle in a three dimensional rectangular infinite potential well. b) Calculate the lowest energy allowed for the electron in this case.

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 \(a / 4\)

Determine the lowest 3 energies of the wave function of a proton in a box of width \(1.0 \cdot 10^{-10} \mathrm{~m}\).
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