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Chapter 40: Problem 7

An electron is in a one-dimensional box. When the electron is in its ground state, the longest-wavelength photon it can absorb is \(420 \mathrm{nm} .\) What is the next longest-wavelength photon it can absorb, again starting in the ground state?

Short Answer

Expert verified
The wavelength of the next longest-wavelength photon that the electron in its ground state can absorb, can be calculated with the aforementioned steps utilizing quantum mechanics concepts, the Planck-Einstein relation and the formula for energy levels in an infinite potential well.

Step by step solution

01

Convert the wavelength into energy

Use the Planck-Einstein relation to convert the initial wavelength (420 nm) into energy, which is \(E_1=h*c/\lambda_1\). Using \(h = 6.626*10^{-34} Js\) (Planck's constant) and \(c = 3.00*10^8 m/s\) (the speed of light), transform the wavelength (位) from nanometers to meters before substituting the values in the equation.
02

Calculate the energy difference between first and second quantum states

The energy levels in an infinite potential well are given by \(E_n = n^2 * h^2 / 8mL^2\), where n is the quantum number, m is the mass of the electron, and L is the length of the well. The difference in energy when the electron jumps from the first to the second energy level is \(\Delta E = E_2 - E_1 = 3E_1 - E_1 = 2E_1\). This is the energy of the photon that must be absorbed for the electron to transition from the ground state (n=1) to the next level (n=2). The energy \(E_1\) was calculated in step 1.
03

Find the wavelength of the second photon

Now, find the wavelength of a photon with this new energy difference, using the rearranged Planck-Einstein relation, \(\lambda_2=h*c/\Delta E\), where \(\Delta E\) was calculated in step 2. This will give the new wavelength of the photon which the electron can absorb to jump from the ground state to the second energy level.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Infinite Potential Well
Imagine a tiny enclosure where particles like electrons can be found. This is what we refer to as an "infinite potential well". It is a simplified model used in quantum mechanics to help understand how electrons behave in confined spaces. Think of it like a box with infinitely high walls that an electron cannot escape from.

In this model, electrons can only exist in certain energy states. The energy levels are quantized, which means they can only take on specific values. This is due to the boundary conditions in the well鈥攅ssentially, there are certain rules that the electron's wave function must follow. Unlike classical physics, where energy can vary smoothly, quantum mechanics dictates discrete energy levels.

The concept of an infinite potential well is crucial in understanding many quantum systems, including how electrons behave in atoms or semiconductors. By studying these models, we learn about how particles move and transition between different energy levels.
Energy Levels
Energy levels in a quantum system like an infinite potential well are akin to the steps on a staircase. Each step represents a possible energy state an electron can have. When an electron is "on" a certain energy level, it has a specific amount of energy and a corresponding wave pattern.

The energy levels are calculated using the formula: \[E_n = \frac{n^2 h^2}{8mL^2}\] Where:
  • \(E_n\) is the energy at the n-th level.
  • \(n\) is the quantum number (1, 2, 3,...).
  • \(h\) is Planck's constant.
  • \(m\) is the mass of the electron.
  • \(L\) is the length of the potential well.
The ground state, or the lowest energy level, is when \(n=1\). As \(n\) increases, so does the energy. When an electron jumps from one energy level to another, it absorbs or emits a photon corresponding to the energy difference between these levels.

Understanding energy levels is essential in predicting how electrons will behave in different situations, such as absorption of light or emission of radiation.
Planck-Einstein Relation
The link between the energy of a photon and its wavelength is elegantly given by the Planck-Einstein relation. This relation is foundational in understanding how light interacts with matter in quantum mechanics.

The Planck-Einstein relation is expressed as:\[E = \frac{hc}{\lambda}\] Where:
  • \(E\) is the energy of the photon.
  • \(h\) is Planck鈥檚 constant \((6.626 \times 10^{-34} \text{ Js})\).
  • \(c\) is the speed of light \((3.00 \times 10^8 \text{ m/s})\).
  • \(\lambda\) is the wavelength of the light.
By using this equation, you can calculate the energy of a photon when you know its wavelength, and vice versa.

This relationship is central to quantum mechanics and is used in a wide array of applications, from understanding the color of stars to the operation of lasers. It helps explain why electrons absorb specific wavelengths of light and how they transition between different energy states.

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Most popular questions from this chapter

When a hydrogen atom undergoes a transition from the \(n=2\) to the \(n=1\) level, a photon with \(\lambda=122 \mathrm{nm}\) is emitted. (a) If the atom is modeled as an electron in a one-dimensional box, what is the width of the box in order for the \(n=2\) to \(n=1\) transition to correspond to emission of a photon of this energy? (b) For a box with the width calculated in part (a), what is the ground-state energy? How does this correspond to the ground-state energy of a hydrogen atom? (c) Do you think a one-dimensional box is a good model for a hydrogen atom? Explain. (Hint: Compare the spacing between adjacent energy levels as a function of \(n .)\)

A particle is confined to move on a circle with radius but is otherwise free. We can parameterize points on this circle by using the distance \(x\) from a reference point or by using the angle \(\theta=x / R .\) since \(x=0\) and \(x=2 \pi R\) describe the same point, the wave function must satisfy \(\psi(x)=\psi(x+2 \pi R)\) and \(\psi^{\prime}(x)=\psi^{\prime}(x+2 \pi R)\) (a) Solve the free-particle time-independent Schr枚dinger equation subject to these boundary conditions. You should find solutions \(\psi_{n}^{\pm}\), where \(n\) is a positive integer and where lower \(n\) corresponds to lower energy. Express your solutions in terms of \(\theta\) using unspecified normalization constants \(A_{n}^{+}\) and \(A_{n}^{-}\) corresponding, respectively, to modes that move "counterclockwise" toward higher \(x\) and "clockwise" toward lower \(x\). (b) Normalize these functions to determine \(A_{n}^{\pm}\). (c) What are the energy levels \(E_{n} ?\) (d) Write the time-dependent wave functions \(\Psi_{n}^{\pm}(x, t)\) corresponding to \(\Psi_{n}^{\pm}(x) .\) Use the symbol \(\omega\) for \(E_{1} / \hbar .\) (e) Consider the nonstationary state defined at \(t=0\) by \(\Psi(x, 0)=\frac{1}{\sqrt{2}}\left[\Psi_{1}^{+}(x)+\Psi_{2}^{+}(x)\right] .\) Determine the probability density \(|\Psi(x, t)|^{2}\) in terms of \(R, \omega,\) and \(t .\) Simplify your result using the identity \(1+\cos \alpha=2 \cos ^{2}(\alpha / 2) .\) (f) With what angular speed does the density peak move around the circle? (g) If the particle is an electron and the radius is the Bohr radius, then with what speed does its probability peak move?

A particle moving in one dimension (the \(x\) -axis) is described by the wave function $$ \psi(x)=\left\\{\begin{array}{ll} A e^{-b x}, & \text { for } x \geq 0 \\ A e^{b x}, & \text { for } x<0 \end{array}\right. $$ where \(b=2.00 \mathrm{~m}^{-1}, A>0,\) and the \(+x\) -axis points toward the right. (a) Determine \(A\) so that the wave function is normalized. (b) Sketch the graph of the wave function. (c) Find the probability of finding this particle in each of the following regions: (i) within \(50.0 \mathrm{~cm}\) of the origin, (ii) on the left side of the origin (can you first guess the answer by looking at the graph of the wave function?), (iii) between \(x=0.500 \mathrm{~m}\) and \(x=1.00 \mathrm{~m}\).

(a) Find the excitation energy from the ground level to the third excited level for an electron confined to a box of width \(0.360 \mathrm{nm}\). (b) The electron makes a transition from the \(n=1\) to \(n=4\) level by absorbing a photon. Calculate the wavelength of this photon.

When low-energy electrons pass through an ionized gas, electrons of certain energies pass through the gas as if the gas atoms weren't there and thus have transmission coefficients (tunneling probabilities) \(T\) equal to unity. The gas ions can be modeled approximately as a rectangular barrier. The value of \(T=1\) occurs when an integral or half-integral number of de Broglie wavelengths of the electron as it passes over the barrier equal the width \(L\) of the barrier. You are planning an experiment to measure this effect. To assist you in designing the necessary apparatus, you estimate the electron energies \(E\) that will result in \(T=1\). You assume a barrier height of \(10 \mathrm{eV}\) and a width of \(1.8 \times 10^{-10} \mathrm{~m} .\) Calculate the three lowest values of \(E\) for which \(T=1\)
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