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Wize University Chemistry Textbook > Early Atomic Theory to Quantum Theory

Standing Waves

Heisenberg Uncertainty PrincipalPrevious Section
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Standing Waves & Particle in a 1-D Box


Standing Waves

Definition: A wave which is confined between two locations in space.

Node: The (non-end) points where a wave undergoes no displacement.

  • Because electrons display wave-like characteristics, they can be approximated as standing waves.
  • Examining the outcomes from a 1-D particle in a box will serve as an illustration for the 3-D “boxes” that electrons occupy, called orbitals.
  • The confinement, as shown in figure 3.4, necessitates that only certain wavelengths are permitted.

λ=(2L)1n\lambda=(2L)\frac{1}{n}


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Wavefunctions


  • The standing waves shown above can be described by mathematical functions, or, wavefunctions
ψn(x)=2Lsin(nπxL)\psi_n(x)=\sqrt{\frac{2}{L}}sin\Big(\frac{n\pi x}{L}\Big)

  • The sin wave provides the oscillation between positive and negative values
  • L is the length of the box
  • The 2L\sqrt{\frac{2}{L}} !term is a normalization term and you are NOT required to understand what it does (for the purpose of this course).


  • The probability of finding a particle at a given position is found by squaring the wavefunction and is denoted ψ2\psi^2

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Kinetic Energy and Quantization


  • When we think about kinetic energy in the macroscopic world, it seems fairly intuitive that kinetic energy could be any value and varies linearly with velocity

  • In the world of subatomic particles, the quantization of wavelength results in a quantization of other properties like kinetic energy

  • Using the classical equation for kinetic energy and momentum, we can add in the equation of de Broglie as well as the wavelength of a particle in one-dimensional box.This allows us to express kinetic energy in terms of n and L as shown below:

Ek=n2h28mL2\textbf{E}_{\textbf{k}}=\frac{\textbf{n}^{\mathbf{2}}\textbf{h}^{\mathbf{2}}}{\mathbf{8}\textbf{mL}^{\mathbf{2}}}

  • n represents which wavefunction we are looking at (n=1,2,3,4...)
  • m is the mass of the particle, typically an electron.
  • L is the length of the box
  • Note that this is the equation for kinetic energy of a particle in a 1-D box





What is the wavelength of a n = 4 standing wave confined to a 210nm length?

λ= 2L(1n)=2(210 nm)(14)=105 nm\lambda=\ 2L\left(\frac{1}{n}\right)=2\left(210\ nm\right)\left(\frac{1}{4}\right)=105\ nm

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Complete the following sentence with the choice that best describes a standing wave

A standing wave is a wave where,

a) The nodes to not move along the length of the wave
b) The amplitude of the wave is the same at each maxima
c) The end-points are stationary
d) An electron is in the wave
e) The wave travels above the speed of light

A standing wave is defined as a wave with fixed endpoints, C. A and B are correct but they do not define a standing wave. D and E make no sense.

C
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Use the graph below to answer the following questions:


1. Which wave has the lowest energy?
D
Solution:
Energy is inversely proportional to wavelength, longest wavelength is D (400 nm)
2. What is the wavelength of wave A?
200 nm
Solution:
Wave A does two complete cycles over 400 nm so, 400nm / 2 = 200m
3. If wave B represents a photon from the UV region, what region of the electromagnetic spectrum would you find wave C?
UV
Solution:
B and C have the same frequency (and wavelength) so they are from the same region
4. Which wave has the lowest amplitude?
B
Solution:
The smallest vertical distance is covered by wave B. ie. it's the shortest wave