Quantum Physics

Step 1: The expression represents the de Broglie wavelength, which describes the wave nature of particles. Step 2: The expression represents the particle's momentum derived from its wavelength.

Matter Waves:
Matter waves, also called de Broglie waves, are the waves associated with particles that exhibit both wave-like and particle-like properties. According to de Broglie's hypothesis, every moving particle, such as an electron or a proton, behaves like a wave and has an associated wavelength. This concept was proposed by Louis de Broglie in 1924 and is known as wave-particle duality. The de Broglie wavelength of a particle with momentum is given by the following formula: where: - is Planck's constant ( ), - is the momentum of the particle ( , where is the mass and is the velocity of the particle). Dual Nature of Particles:
The concept of wave-particle duality was experimentally verified by the Davisson-Germer experiment, which demonstrated that electrons exhibit diffraction patterns, similar to light waves, when passed through a crystal lattice. Final Answer: The formula for de Broglie wavelength is: The experiment that shows the dual nature of particles is the Davisson-Germer experiment.

Step 1: Understanding the Concept:
In quantum mechanics, light is described as being composed of discrete packets of energy called photons. The energy of a single photon is related to the frequency and wavelength of the corresponding electromagnetic wave.

Step 2: Key Formula or Approach:
The derivation involves two fundamental equations: \begin{enumerate} \item The Planck-Einstein relation, which gives the energy of a photon ( ) in terms of its frequency ( ): where is Planck's constant.
\item The wave equation, which relates the speed of light ( ), its frequency ( ), and its wavelength ( ): \end{enumerate}

Step 3: Detailed Explanation:
Our goal is to express the energy in terms of and . The first equation has but uses , not . We need to replace using the second equation.
From the wave equation, we can express frequency as: Now, substitute this expression for into the Planck-Einstein relation: This is the required equation.

Step 4: Final Answer:
The equation for the energy of a photon ( ) in terms of Planck's constant ( ), the speed of light ( ), and wavelength ( ) is:

The de Broglie wavelength ( ) of a particle is related to its momentum ( ) by the equation:
Where:
- is the de Broglie wavelength,
- is Planck's constant ( ),
- is the momentum of the particle.
Since momentum is given by , where is the mass and is the velocity of the particle, we can write:
Now, consider a proton and an -particle (helium nucleus) accelerated by the same potential difference. The kinetic energy acquired by a particle when accelerated by a potential is given by:
Where is the charge of the particle and is the potential. For a proton, and for an -particle, . From this, we can find the velocity of each particle and subsequently the momentum.
For the proton, the velocity is:
For the -particle, the velocity is:
Now, the de Broglie wavelength for the proton ( ) and the -particle ( ) are:
Finally, the ratio of their de Broglie wavelengths is:
Since the mass of a proton is approximately 1836 times the mass of an electron, and the mass of an -particle , we get:
Thus, the ratio of the de Broglie wavelengths of the proton and -particle is approximately 2.83.

The de Broglie wavelength of a particle is given by the formula: Where: - is Planck's constant ( ), - is the momentum of the particle.
The momentum of a particle accelerated by a potential difference is given by: Where: - is the mass of the particle,
- is the charge of the particle ( ),
- is the potential difference.
Now, we calculate the momentum for both the proton and the -particle.
1. For the proton:
- Mass of proton ,
- Charge of proton .
The momentum of the proton is: 2. For the -particle:
- Mass of -particle ,
- Charge of -particle .
The momentum of the -particle is: 3. Ratio of the de Broglie wavelengths:}
Since the de Broglie wavelength is inversely proportional to the momentum, the ratio of the wavelengths is: Simplifying the ratio: Thus, the ratio of the de Broglie wavelengths is approximately 2.83.

The de-Broglie equation relates the wavelength of a particle to its momentum. It shows that matter exhibits wave-like properties, just like light. The equation is given by: Where:
- is the de-Broglie wavelength, which is the wavelength associated with the particle,
- is Planck's constant, with a value of ,
- is the momentum of the particle. Momentum is given by the product of mass and velocity ( ), so the equation becomes: This shows that the wavelength is inversely proportional to both the mass and the velocity of the particle. The de-Broglie wavelength is especially significant for subatomic particles, such as electrons, where the wavelength becomes comparable to atomic distances.
For macroscopic objects, the de-Broglie wavelength is incredibly small, so it does not have noticeable effects. However, for very small particles like electrons, neutrons, and protons, the wave-like nature is significant, and it explains phenomena like electron diffraction.
In conclusion, de-Broglie postulated that all matter has an associated wavelength, and the smaller the mass of the particle, the larger its wavelength at a given velocity. The wavelength of an object moving at high speeds (close to the speed of light) becomes comparable to the wavelength of light itself. This wave-particle duality is a fundamental concept in quantum mechanics.

Step 1: Understanding the Concept:
The de Broglie hypothesis proposes that all matter, such as electrons and protons, exhibits wave-like properties. A particle with momentum has an associated wavelength , known as the de Broglie wavelength. Kinetic energy (K) is the energy a particle possesses due to its motion. The task is to derive an expression for the wavelength in terms of the particle's kinetic energy .

Step 2: Key Formula or Approach:
The de Broglie wavelength is fundamentally defined as: where is Planck's constant and is the momentum of the particle ( ).
The classical kinetic energy (K) of a particle of mass is given by . This can be related to momentum by the formula:

Step 3: Detailed Explanation:
Our goal is to substitute momentum in the de Broglie equation with an expression involving kinetic energy . We start with the kinetic energy formula and solve for : Multiplying both sides by : Taking the square root of both sides gives the momentum: Now, we substitute this expression for into the de Broglie wavelength formula: This equation gives the de Broglie wavelength directly from the particle's kinetic energy and mass.

Step 4: Final Answer:
The de Broglie wavelength ( ) in terms of kinetic energy ( ) for a particle of mass is given by the expression: