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Chapter 39: Problem 10

An electron is trapped in a one-dimensional infinite potential well. For what (a) higher quantum number and (b) lower quantum number is the corresponding energy difference equal to the energy difference \(\Delta E_{43}\) between the levels \(n=4\) and \(n=3 ?\) (c) Show that no pair of adjacent levels has an energy difference equal to \(2 \Delta E_{43}\).

Short Answer

Expert verified
Quantum difference for higher/lower: details misalign intrinsically while adjacently doubling doesn't align pair of integer.

Step by step solution

01

Understand the Energy Levels in an Infinite Potential Well

The energy levels of an electron in a one-dimensional infinite potential well are given by the equation: \[ E_n = \frac{n^2 h^2}{8mL^2} \] where \(n\) is the quantum number, \(h\) is Planck's constant, \(m\) is the mass of the electron, and \(L\) is the length of the well.
02

Calculate the Energy Difference \\(\Delta E_{43} \\\)

The energy difference between levels \(n=4\) and \(n=3\) is given by: \[ \Delta E_{43} = E_4 - E_3 = \frac{16 h^2}{8mL^2} - \frac{9 h^2}{8mL^2} = \frac{7 h^2}{8mL^2} \] This will be our reference energy difference.
03

Determine the Higher Quantum Number Condition

To find a higher quantum number \(n\), the energy difference \(\Delta E_{n(n-1)}\) should equal \(\Delta E_{43}\), thus: \[ \Delta E_{n(n-1)} = E_n - E_{n-1} = \frac{n^2h^2}{8mL^2} - \frac{(n-1)^2 h^2}{8mL^2} = \Delta E_{43} = \frac{7h^2}{8mL^2} \] Solve for \(n\): \[ n^2 - (n-1)^2 = 7 \] \[ n^2 - (n^2 - 2n + 1) = 7 \] \[ 2n - 1 = 7 \] \[ 2n = 8 \] \[ n = 4 \] So, \(n = 4\), which means our assumption was too restrictive as it's an equality check.
04

Determine the Lower Quantum Number Condition

Next, find a lower pair \(n, n-1\) so: \[ \Delta E_{(n+1)n} = \Delta E_{43} \] We need \[ \Delta E_{21} \] below, check the capability that lower \(n\) does fulfill it is generally not possible to have smaller numbers with equal condition since \(7\) difference aligns usually for certain factor ensuring it equivalent above \(3\).
05

Verify Doubling the Energy Difference \(2\Delta E_{43}\) for Adjacent Levels

Consider \( \Delta E_{(n+1)n} = 2\Delta E_{43} \). This is: \[ E_{n+1} - E_n = 2\Delta E_{43} = 2\left(\frac{7h^2}{8mL^2}\right) \] Substituting \(E_{n+1}\) and \(E_n\) expressions: \[ \frac{(n+1)^2 h^2}{8mL^2} - \frac{n^2 h^2}{8mL^2} = \frac{14h^2}{8mL^2} \] Simplifying: \[ (n+1)^2 - n^2 = 14 \] \[ n^2 + 2n + 1 - n^2 = 14 \] \[ 2n + 1 = 14 \] \[ 2n = 13 \] \[ n = 6.5 \] This is an impossibility (non-integer), confirming no pair of adjacent levels has an energy difference equal to \(2 \Delta E_{43}\).

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Key Concepts

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

Quantum Numbers
In quantum mechanics, quantum numbers are essential for describing the quantum state of a particle, such as an electron in an atom or a molecule. They help identify the quantized energy levels in which the electron can exist. Quantum numbers usually have specific rules:
  • The principal quantum number (\(n\)) indicates the energy level and relative size of the electron cloud. It鈥檚 usually a positive integer.
  • As \(n\) increases, the energy and size of the electron cloud increase as well.
  • Quantum numbers are discrete, meaning electrons exist in set quantized states.
The exercise here explores how changes in quantum numbers affect the energy difference between states. For instance, calculating the energy difference \(\Delta E_{43}\) involves determining the energy between the 4th and 3rd quantum levels, highlighting how altering quantum numbers alters energy.
Infinite Potential Well
An infinite potential well is a fundamental model in quantum mechanics. It's also called a particle in a box.
  • The concept represents a particle that is restricted to move in a confined space, like an electron constrained between two non-penetrable walls.
  • In this model, the potential energy inside the well is zero, while it's infinite outside. Hence, particles cannot escape this region.
  • The walls create boundary conditions that quantize particle energies, forcing them to take on only specific values.
The energy of the particle, like an electron, is determined by the quantum number and the dimensions of the well. For a one-dimensional potential well, solving the Schr枚dinger equation gives us quantized energy levels \(E_n\). This means that particles like electrons in these wells can only have particular energy levels, derived from the equation \(E_n = \frac{n^2 h^2}{8mL^2}\). Here, \(L\) represents the length of the well, and \(m\) is the particle's mass.
Energy Levels
In quantum mechanics, energy levels are quantized, discrete values that a system like an atom, molecule, or particle can possess:
  • Energy levels are determined by solving the Schr枚dinger equation for the system under study, as for a particle in a potential well.
  • Levels are often indicated by quantum numbers, \(n\) representing the principal quantum number in our example.
  • The energy difference between two levels, like \(\Delta E_{43}\), can be calculated by knowing the energies at each level.
For an infinite potential well, energy levels are given by \(E_n = \frac{n^2 h^2}{8mL^2}\). The task involves comparing these energy differences, showing that only certain changes in quantum numbers (e.g., moving from \(n=3\) to \(n=4\)) produce a specific energy change, achievable only at defined quantum levels.
Planck's Constant
Planck's constant is a pivotal element in the realm of quantum mechanics:
  • Symbolized as \(h\), it is a fundamental constant that relates the energy of a photon to the frequency of its electromagnetic wave.
  • The formula connecting energy and frequency is \(E = hu\), where \(E\) is energy and \(u\) is frequency.
  • Planck's constant appears in formulas calculating energy levels, such as \(E_n = \frac{n^2 h^2}{8mL^2}\) for the infinite potential well.
This constant helps define the scale of quantum effects with such quantized systems. It acts as a bridge between quantum and classical worlds and is crucial in understanding how quantization affects energy levels in systems like the infinite potential well, highlighting the relationship between quantum number changes and energy level differences.

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Most popular questions from this chapter

What are the (a) energy, (b) magnitude of the momentum, and (c) wavelength of the photon emitted when a hydrogen atom undergoes a transition from a state with \(n=3\) to a state with \(n=1 ?\)

A proton is confined to a one-dimensional infinite potential well \(100 \mathrm{pm}\) wide. What is its ground-state energy?

An electron is trapped in a one-dimensional infinite potential well that is \(100 \mathrm{pm}\) wide; the electron is in its ground state. What is the probability that you can detect the electron in an interval of width \(\Delta x=5.0 \mathrm{pm}\) centered at \(x=(\) a) 25 \(\mathrm{pm}\), (b) \(50 \mathrm{pm}\), and \((\mathrm{c}) 90 \mathrm{pm} ?\) (Hint: The interval \(\Delta x\) is so narrow that you can take the probability density to be constant within it.)

The ground-state energy of an electron trapped in a onedimensional infinite potential well is \(2.6 \mathrm{eV}\). What will this quantity be if the width of the potential well is doubled?

An electron, trapped in a one-dimensional infinite potential well \(250 \mathrm{pm}\) wide, is in its ground state. How much energy must it absorb if it is to jump up to the state with \(n=4 ?\)
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