De Broglie Compact with PYQ

The De Broglie Hypothesis and Wave Mechanics

The De Broglie hypothesis extended wave-particle duality to material bodies.

Hypothesis: A moving body behaves in certain ways as though it has a wave nature.
Photon Momentum: A photon of light with frequency $ν$ has momentum $p = h ν / c$ . Since $c = ν λ$ , the wavelength of a photon is specified by its momentum as $p = h/ λ$ .
De Broglie Wavelength (Generalization): This relation is generalized to apply to material particles as well.
- Momentum of a particle: $p = m v$ .
- De Broglie Wavelength: $λ = \frac{h}{p} = \frac{h}{mv}$ (Eq. 2).
- Note: $m$ in this equation is the relativistic mass, $m = m_{0} / 1 - v^{2} / c^{2}$ .
- Relationship: The greater the particle’s momentum, the shorter its wavelength.
Duality: The wave and particle aspects of moving bodies can never be observed at the same time. Which property is more conspicuous depends on how the De Broglie wavelength compares with the dimensions of the body or what it interacts with.
Verification: Equation (2) was later verified by experiments involving the diffraction of electrons by crystals.

Numerical PYQ	Year(s)	Hint / Formula / Solution (If external, noted)
An electron has de-Broglie wavelength of 2 PM. Find its Kinetic energy and the phase & group velocities of its de-Broglie waves.	2021	Formulas (Partially External): 1. Kinetic Energy ( $T$ ): For non-relativistic speeds, $T = p^{2} / (2 m) = h^{2} / (2 m λ^{2})$ . Use $λ = 2 \times 1 0^{- 12}$ m. 2. Particle Velocity ( $v$ ): $v = 2 T / m$ . 3. Phase Velocity ( $w$ ): $w = c^{2} /v$ . 4. Group Velocity ( $v_{g}$ ): $v_{g} = v$ (the particle velocity) (Not in source slides).
Calculate the wavelength associated with a thermal neutron of energy 0.025 eV.	2022	Formula (External): Since $E$ (Kinetic Energy) is given, use $\mathbf{\lambda = \frac{h}{\sqrt{2m_n E}}$ . Remember to convert $E$ from eV to Joules ( $E = 0.025 \times 1.6 \times 1 0^{- 19}$ J). $m_{n}$ (mass of neutron) is an external constant.

The variations that constitute matter waves are described by the wave function, $Ψ$ (psi).

Wave Function ( $Ψ$ ): The value of $Ψ$ at a point $(x, y, z)$ at time $t$ is related to the likelihood of finding the body there at that time.
Lack of Direct Physical Significance: $Ψ$ itself has no direct physical significance. Since probability must lie between 0 and 1, and $Ψ$ (being an amplitude) can be negative, $Ψ$ cannot be an observable quantity.
Probability Density: The objection is resolved using $∣Ψ ∣^{2}$ , the square of the absolute value of the wave function, known as probability density.
- The probability of finding the body described by $Ψ$ is proportional to the value of $∣Ψ ∣^{2}$ .
- A large $∣Ψ ∣^{2}$ means a strong possibility of the body’s presence.
Particle vs. Likelihood: When an experiment detects the particle (e.g., an electron), a whole particle is found; the wave function describes the likelihood of where it will be found.

The velocity of the De Broglie waves, denoted $w$ (the Phase Velocity), can be derived using wave and quantum relations.

Wave Formula: $w = ν λ$ .
Frequency ( $ν$ ): Derived from quantum energy $E = h ν$ , so $ν = E / h$ .
Relativistic Energy: Using $E = m c^{2}$ , we get $ν = m c^{2} / h$ .
Phase Velocity ( $w$ ): Substituting $ν$ and $λ = h / (m v)$ into $w = ν λ$ : $w = \frac{c ^{2}}{v}$ (Eq. 3).
Interpretation: Since particle velocity ( $v$ ) must be less than the speed of light ( $c$ ), the calculated De Broglie waves always travel faster than light ( $w > c$ ). This necessitates distinguishing $w$ (Phase Velocity) from the Group Velocity.