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Wize University Chemistry Textbook > Early Atomic Theory to Quantum Theory

Wave-Particle Duality & the de Broglie Equation

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Wave-Particle Duality

Light was originally thoughy of as only having wave-like properties, but was later shown to have particle-like properties as well.

Light as Waves


  • When light was shown at a slit, interference patterns were recorded
  • Interference patterns are a property of waves


Light as Particles



  • Einstein showed that light can also be thought of as particles since they are like packages of exact amounts of energy
  • The experiment was the photoelectric effect and it showed that there is wave-particle duality (light has both properties of waves and particles!).


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What About Electrons?

Louis de Broglie thought that if light has both properties of waves and particles, electrons might also have properties of particles and waves.

He determined the following equation to explain the wave-nature of particles:

λ=hmv=hp\boxed{\lambda=\frac{h}{mv}=\frac{h}{p}}
De Broglie Equation

λ is the de Broglie wavelength of the small particle (m)
h is Planck's constant= 6.626 x10-34 Js (or 6.626x10-34 kg m2/s)
m is the mass of the particle (kg)
v is the velocity of the particle (m/s)
p is the momentum of the particle (p=mv)
Watch Out!
The v in this equation is not frequency (v, Hz), it is velocity (v, m/s).

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Light acting as a wave:


Young's double slit light experiment
Photo by Rice University / CC BY


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Could electrons also act as waves?


Photo by Johannes Kalliauer / CC BY

In summary

All waves can behave like particles and all particles can behave like waves! The wave-particle duality applies to light and particles!
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Example: Wavelength of an Accelerated Neutron

At CERN, a neutron is accelerated to 2.4 x 108 m/s in a particle accelerator. The mass of the neutron is 1.675 x 10-27 kg, what is it's wavelength? (h is 6.626x10-34 kg m2/s)

a) 1.14 x 10-15 m
b) 2.77 x 10-14 m
c) 1.65 x 10-15 m
d) 3.53 x 10-14 m

We are being asked for the wavelength of a particle, we must use the de Broglie equation!
λ=hmv=6.626×10−34Js(1.675×10−27kg)(2.4×108 m/s)=1.65×10−15m\lambda=\dfrac{h}{mv}=\dfrac{6.626\times 10^{-34} Js}{(1.675\times 10^{-27} kg)(2.4\times 10 ^8\ m/s)}=1.65\times 10^{-15}m
Answer is C

Practice: Two-Slit Experiment

If you shoot a beam of electrons at a piece of metal with two small slits in it, a diffraction pattern appears on the detector on the otherside. This is often referred to as a two slit experiment. This result shows that,

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Practice: Wave-Particle Duality


Nolan Ryan, a fastball pitcher from the 1970’s, was recorded throwing a four seam fastball around 108mph (174km/h). A MLB baseball weighs approximately 145g. What is the wavelength of the baseball?


v=174kmh×1000mkm×1h3600s=48.3ms−1v=\frac{174km}{h}\times\frac{1000m}{km}\times\frac{1h}{3600s}=48.3ms^{-1}

λ=hmv=6.626 ×10−34kgm2s−1(0.145kg)(48.3ms−1)=9.46 ×10−35m\lambda=\frac{h}{mv}=\frac{6.626\ \times10^{-34}kgm^2s^{-1}}{\left(0.145kg\right)\left(48.3ms^{-1}\right)}=9.46\ \times10^{-35}m

Extra Practice
Light Energy
The energy content of a particular form of light is 1.572 J per mole of photons. What is the wavelength of this light? (Choose the closest answer.)
Electromagnetic radiation: Wave energy
A laser emits light at a wavelength of 710 nm.
What is the frequency of this light?
What is the de Broglie wavelength of a 20 kg dog? (Note: 1 J  = 1 kg m2s21\ J\ \ =\ 1\ kg\ \frac{m^2}{s^2})
If a photon was able to remove an electron from a metal plate that had a binding energy of 2 x 10-20 J, what was the frequency of the wave?
If the answer has an exponent write it like this 1.23x10^4 (no spaces)
Light and EM Multipart Question
You have a laser that emits light at a wavelength of 710 nm.
Electromagnetic radiation: Wave energy
A laser emits light at a wavelength of 710 nm.
What is the frequency of this light?
Work Function
The work function of metallic cesium is 3.36×10−19 J. What is the threshold wavelength of light required to emit electrons from the surface of metallic cesium?
Light Energy
The energy content of a particular form of light is 1.572 J per mole of photons. What is the wavelength of this light? (Choose the closest answer.)
H atom
A hydrogen atom, initially in its lowest energy level, absorbs a photon with a wavelength of 93.1 nm. What is the final energy level?
H-like system
Which of the following electron transitions for the hydrogen atom is associated with the longest photon wavelength?
de Broglie wavelength
Which of the following objects has the longest de Broglie wavelength? Assume they are all moving at the same speed.
Wave particle duality
The wavelike character of ordinary objects like a thrown baseball cannot be detected because:
Energy of Light
A domestic microwave generates radiation of wave length 9.3 cm (very low energy).
a. What are the energy values per photon and per mole of of photons?
b. Calculate the frequency of this radiation
Water can absorb infrared radiation at a wavelength of1.06×104 m{\small 1.06\times10^4 \text{ m}}. If a 180 g sample of water is irradiated at this wavelength, calculate the number of photons required to raise the temperature by 25.0°C(CH2O=4.184Jg°C){\small \degree \text{C} \left(C_{H_2O} =4.184\frac{J}{g\degree C} \right)}
Electromagnetic radiation: Wave Energy
Calculate the energy of one photon whose wavelength is 589 nm.
How heavy is a particle whose de Broglie wavelength is 1.87 x 10-16 m?
Practice Problem 1: de Broglie Equation
What is the energy associated with the de Broglie wavelength of a 1000 kg car? ((1 J = 1kgm2s2)\left(1\ J\ =\ 1kg\frac{m^2}{s^2}\right)
An emission spectrum for a one-electron species is shown below. All the lines correspond to transitions ending at the energy level n = 3. Line B has a wavelength of 165 nm. Identify the species that produced this spectrum.
Identify the species in the spectrum above
If a photon was able to remove an electron from a metal plate that had a binding energy of 2 x 10-20 J, what was the frequency of the wave?
Wave Particle Duality: de Broglie's Postulate
Calculate the de Broglie wavelength for the following objects. For each one decide whether they are best treated as classical or quantum mechanical particles
A proton moving at 1% of the speed of light
How heavy is a particle whose de Broglie wavelength is 1.87 x 10-16 m?
What is the de Broglie wavelength of a 20 kg dog? (Note: 1 J  = 1 kg m2s21\ J\ \ =\ 1\ kg\ \frac{m^2}{s^2})
Wave Particle Duality and the Heisenberg Uncertainty Principle
A beam of electrons is accelerated towards a thin metal foil and diffracts to give the exact same pattern as a blue laser with wavelength of 480nm. (me = 9.11 x 10-31 kg)
a) What speed were the electrons accelerated to?'
b) If we measure the speed to a precision of ± 1% what is the minimum uncertainty in the position of the electron?
A proton (mp = 1.6726 x 10-27 kg) was found to have a de Broglie wavelength of 522 nm. How fast was the particle moving?
Quantum Model: de Broglie's Equation
What is the energy associated with the de Broglie wavelength of a 1000 kg car? ((1 J = 1 kgm2s2)\left(1\ J\ =\ 1\ kg\frac{m^2}{s^2}\right)
Wave-Particle Duality: Wave Energy
Scientists in the lab were preparing a new chemical reaction. Upon mixing the chemicals, the reaction began to emit a great deal of light. The scientists were able to determine that each photon of light was releasing 8.32 x 10-19 J of energy. What is the wavelength of this light? What is the frequency?
Wave Particle Duality: de Broglie's Postulate
The electrons used in electron microscopy have a velocity of 70% of the speed of light. What is their wavelength in nm?
The reason that we can't measure the position and velocity of a particle to infinite precision is?
What wavelength of light corresponds to a transition from a 6p orbital to a 1s orbital in hydrogen?
Report your answer to three significant figures in nm. Please do not include units in the answer field.
The experiment that showed electrons can behave like waves is the:
de Broglie's postulate: Wavelength of Lance Armstrong
During the tour de France in 2010, Lance Armstrong reached a speed of 57.8km/h descending one of the mountains. If we treat Lance and his bike as a single particle with a weight of 100kg what is the wavelength of Lance Armstrong?
Young's Double-Slit Experiment
If you shoot a beam of electrons at a piece of metal with two small slits in it, a diffraction pattern appears on the detector on the otherside. This is often refereed to as a two slit experiment. This result shows that,
What is the wavelength of an electron traveling at 5% the speed of light?