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An expression for de Broglie wavelength of matter waves is:

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λ = \frac{h}{p}

λ = \frac{h}{2 p}

λ = \frac{h}{4 p}

None of these

answer is A.

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Detailed Solution

Concept- We'll start by comparing Einstein's equation connecting matter and energy with Planck's equation, with the assumption that the energies in both cases are equal.
Formulas Used:

and

De Broglie obtained his equation using well-known ideas by making the following substitutions:
Einstein's famous equation connecting matter and energy states that

where

denotes energy,

denotes mass and

denotes the speed of light
Using Planck's theory which states every quantum of a wave has a discrete amount of energy given by Planck's equation

where

denotes the energy,

denotes the planck's constant and

denotes the frequency
The value of the planck's constant is given by

Joule-sec
Because physical particles do not travel at the speed of light, de Broglie substituted v for the particle's velocity for the speed of light (c)
We can solve the equation (1) by replacing c with v.

De Broglie theorised that because particles and waves had the same properties, their energy would be equal.
The LHS of both equations (2) and (3) is clearly the same (i.e., energy E). As a result, the RHS of both equations may be equaled and will be equal.

We can write anything based on the relationship between the wavelength, velocity, and frequency of any wave.

By substituting

in equation (4), we get

As we know that the momentum

is the product of mass

and the velocity

i.e.,

We get by inserting p = mv in equation (5).

where

denotes de Broglie wavelength of matter waves,

is planck's constant (

Joule-sec) and

denotes the momentum of the matter wave.
The above equation is the necessary expression for the de Broglie wavelength of matter waves.
Hence, the correct answer is option 1.

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