De Broglie Relationship: Definition, Derivation and Sample Questions

De Broglie relationship determines that similar to light, matter also exhibits wave-like and particle-like properties. This nature has been described as dual behaviour of matter. Based on his observations, de Broglie derived a relationship between momentum of matter and wavelength. This relationship is termed as the De Broglie Relationship.

Keyterms: Matter, Momentum, Wavelength, Atoms, Spectrum, Magnetic fields, Electric fields, Bohr’s model, Atomic model

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What is the De Broglie Relationship?

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Many concepts related to the spectrum of various atoms and the splitting of spectral lines under magnetic and electric fields were not explained by Bohr's model. To address the flaws in Bohr's atomic model, efforts were undertaken to build a more comprehensive atomic model.

The examination of dual behaviour of matter was one of those developments that had a profound impact on the formulation. Considering the particle nature, Einstein equation is shown as,

E= mc² …...(1)

Where,

E = energy

m = mass

c = speed of light

Considering the wave nature, the Plank’s equation is given as,

E = hν …... (2)

Where,

E = energy

h = Plank’s constant

ν = frequency

From (1) and (2),

mc² = hν …... (3)

Frequency, ν can be expressed in terms of wavelength, λ as,

ν = c/λ

For a general particle, c can be replaced with the velocity of object, v. Hence, equation (3) can be given as,

mv² = hv/λ ⇒ λ = h/mv

The above equation is known as de Broglie relationship and the wavelength, λ is known as de Broglie wavelength. 

Every object in motion has a wavelike character. The wavelengths associated with ordinary objects are short due to a large mass. Due to this, their wave properties cannot be detected. On the contrary, the wavelengths connected with electrons and other subatomic particles can be experimentally detected.

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Significance of De Broglie Equation

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de Broglie explained that all the objects in motion have a particle nature. Yet, if we look at a moving car or a moving ball, they don’t seem to have particle nature. To explain this further, de Broglie derived the wavelengths of a cricket ball’s electrons. 

De Broglie Wavelength 

1. De Broglie Wavelength for a Cricket Ball

Let’s assume,

Mass of the ball = 150 g (150 x 10^-³ kg),

Velocity = 35 m/s, and

h = 6.626 x 10^-34 Js

Now, putting these values in the equation λ = hmv

λ = 6.626×10⁻³⁴/150×10⁻³×35

On solving, we get,

λBALL = 1.2621 x 10⁻³⁴

m = 1.2621 x 10⁻²⁴ Å

Å is a very small unit, and thus, the value is in the power of 10⁻²⁴. Here onwards, we see that the moving cricket ball is a particle.

Now, the question arises whether this ball has a wave nature or not. Answer will be a big no because the value of λBALL is immeasurable. 

Hence, de Broglie’s theory of wave-particle duality is valid for the moving objects ‘up to’ the size (not equal to the size) of the electrons.

2. De Broglie Wavelength for an Electron

We know that m_e = 9.1 x 10⁻³¹ kg, and v_e = 218 x 10⁶ m/s 

Now, putting these values in the equation λ = h/mv = 9.1×10⁻³¹/218×10⁶

λ = 3.2 Å 

This value is measurable. Thus, we can say that electrons have wave-particle duality. Also, all the big objects have a wave nature and microscopic objects like electrons have wave-particle nature.

De Broglie Hypothesis: Conclusion 

From de Broglie equation for a material particle, i.e., λ = h/p = h/mv, we conclude that:

If v = 0, then λ = ∞, and,

If v = ∞, then λ = 0

This denotes that waves are associated with the moving material particles only. Therefore, these waves are independent of their charge.

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De Broglie Wavelength of an Electron Derivation

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Let’s say an electron accelerated from resting position through a voltage difference of V volts.

Then,

Gain in KE = ½ mv²

Work done on the electron = eV

So, ½ mv² = eV or v = √2eV/m

We know that λ = h/mv = h/√2meV

On substituting the values:

h = 6.626 x 10⁻³⁴ J/s 

m = 9.1 x 10⁻³¹kg

e = 1.6 x 10⁻¹⁹ C

On solving, we get

λ = 12.27/√VÅ

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De Broglie Equation Derivation and de Broglie Wavelength

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Particles of very low mass travelling at less than the speed of light act like a particle and a wave. de Broglie devised an equation that associates the mass of such smaller particles to their wavelength.

Plank’s quantum theory explains the connection between energy of an electromagnetic wave and its wavelength or frequency.

E = hν = hc/λ……(1)

Einstein associated the energy of particle matter to its mass and velocity, as E = mc²……..(2)

As the smaller particle has a dual nature, and energy is the same, de Broglie equated both these relations for the particle moving with velocity ‘v’ as,

Here, ‘h’ is the Plank’s constant.

This equation connecting a particle's momentum to its wavelength is known as the de Broglie equation, and the wavelength determined using this connection is known as the de Broglie wavelength.

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Things to Remember

• De Broglie wave, also known as matter wave, is any element of a material object's behaviour or characteristics that varies in time or space in accordance with the mathematical equations that describe waves.
• The wavelength of any moving object may be calculated using the de Broglie wave equations.
• The wavelength of an electron increases with reduction in its speed.
• De Broglie established the following relationship between a material particle's wavelength () and momentum (p). λ=h/mv=h/p. Here is the wavelength, and p is the momentum.
• Every moving item has a wave nature. Because of their large masses, the wavelengths associated with common objects are so short that their wave characteristics cannot be determined.

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