Matter Wavelength and Kinetic Energy
For a non-relativistic particle, K = p squared divided by 2m. Substitution into lambda = h/p gives lambda = h divided by the square root of 2mK.
Why this shows up in the exam
Reading wavelength-energy graphs · Finding energy changes from wavelength ratios · Comparing electron wavelength at two kinetic energies
Learn the idea
For fixed non-relativistic mass, de Broglie wavelength varies as the inverse square root of kinetic energy. Kinetic energy grows with the square of momentum, while wavelength falls as the inverse of momentum. Combining these facts makes wavelength fall only as the square root of energy.
🧠 Memory hook: At fixed mass, wavelength follows one over root K.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- lambda = h / sqrt(2 m K) — non-relativistic matter wavelength in terms of kinetic energy
- K lambda² = h²/(2m) — constant relation for particles of one fixed mass
How to approach it
- 1Confirm that the motion is non-relativistic
- 2Write lambda proportional to one over square root of mK
- 3Square wavelength ratios only after arranging the ratio correctly
Common slip-ups that cost marks
- •Using lambda proportional to one over K
- •Comparing different masses with a fixed-mass shortcut
- •Using the non-relativistic expression at very high kinetic energy
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
Photons of energy 5 eV illuminate a metal of work function 2 eV. Find the stopping potential.
More from Dual Nature of Matter and Radiation
Photoelectric Effect
The photoelectric effect describes the emission of electrons from a material when it is exposed to light of sufficient frequency, governed by concepts such as threshold frequency, work function, stopping potential, and the Einstein photoelectric equation.
de Broglie Wavelength and Matter Waves
All matter exhibits wave-like properties, with the de Broglie wavelength inversely proportional to momentum and dependent on factors like velocity, temperature, and particle type.
Photon Properties and Energy-Momentum Relations
Photons are massless particles of light characterized by their energy, frequency, momentum, and charge neutrality, and their interactions obey conservation laws.
Photon Energy and Momentum
For a photon in vacuum, energy is proportional to frequency and momentum is energy divided by c. Frequency and wavelength obey c = nu lambda, so shorter-wavelength photons have larger energy and momentum.
Photon Rate, Power, and Energy Density
For monochromatic radiation, total energy is the number of photons times h nu. Power is energy per unit time, so the photon emission rate equals power divided by single-photon energy.
Radiation Pressure and Photon Momentum Transfer
Radiation force is the rate of photon momentum transfer. For normal incidence on an ideal absorber the pressure is intensity divided by c; for an ideal reflector it is twice that value.