Matter waves, de Broglie relation
- De-Broglie wavelength of an electron orbiting in the n = 2 state of hydrogen atom is close to (Given Bohr radius = 0.052 nm) Answer: 2. 0.67 nm Key idea: Bohr's quantization 2π r_n=nλ (electron wave fits the orbit). Radius of the n=2 orbit: r₂=n²a₀=4×0.052=0.208 nm... 1
- A photon and an electron (mass m) have the same energy E. The ratio ≤ft(λ_photon/λ_electron) of their de Broglie wavelengths is: (c is the speed of light) Answer: 3. c√(2m/E) Key idea: λ=h/p , with different momentum-energy relations for photon and electron. Photon: E=pc⇒λ_ph=hc/E . Electron: E... 1
- The graph which shows the variation of ≤ft(1/λ^2) and its kinetic energy, E is (where λ is de Broglie wavelength of a free particle) : Answer: 1. graph (1) Key idea: For a free particle E=p²/2m and λ=h/p . So E=h²/2mλ² , which gives frac1λ²=2m/h²E . frac1λ² is directly propo... 1
- The graph which shows the variation of the de Broglie wavelength (λ) of a particle and its associated momentum (p) is : Answer: 1. (graph) de Broglie wavelength is λ=h/p , so λ is inversely proportional to p . The graph of λ against p is therefore a rectangula... 1
- An electromagnetic wave of wavelength 'λ' is incident on a photosensitive surface of negligible work function. If 'm' mass is of photoelectron emitted from the surface has de-Broglie wavelength λ_d, then : Answer: 3. λ = (2mc/h)λ_d² Key idea: with negligible work function, the photoelectron's kinetic energy equals the photon energy. hc/λ=p²/2m⇒... 1
- An electron is accelerated from rest through a potential difference of V volt. If the de Broglie wavelength of the electron is 1.227×10^-2 nm, the potential difference is : Answer: 4. 10⁴ V de Broglie wavelength of an electron accelerated through V volts: λ=1.227/sqrt V nm. Given λ=1.227×10⁻² nm: sqrt V=frac1.22... 1
- An electron is accelerated through a potential difference of 10,000 V. Its de Broglie wavelength is, (nearly) : (m_e = 9×10^-31 kg) Answer: 2. 12.2×10⁻¹² m de Broglie wavelength of an electron accelerated through V : λ=h/√(2meV) . Substituting h=6.63×10⁻³⁴ , m=9×10⁻³¹ , e... 1
- An electron of mass m with an initial velocity V=V_0 i (V_0>0) enters an electric field E=-E_0 i (E_0 = constant >0) at t = 0. If λ_0 is its de-Broglie wavelength initially, then its de-Broglie wavelength at time t is Answer: 1. λ₀/(1+eE₀/mV₀t) The electron has charge -e and the field is -E₀hat i , so the force is +eE₀hat i . It accelerates along its motio... 1
- The de-Broglie wavelength of a neutron in thermal equilibrium with heavy water at a temperature T (Kelvin) and mass m, is : Answer: 1. h/√(3mkT) Key idea: a thermal neutron has average kinetic energy 3/2kT , and λ=h/p=h/√(2mK) . λ=h/√(2m·3/2kT)=h/√(3mkT) . 1
- Electrons of mass m with de-Broglie wavelength λ fall on the target in an X-ray tube. The cutoff wavelength (λ_0) of the emitted X-ray is Answer: 4. λ₀=2mcλ²/h The cutoff wavelength corresponds to the whole kinetic energy K of the electron turning into one photon: λ₀=hc/K . de... 1
- An electron of mass m and a photon have same energy E. The ratio of de-Broglie wavelengths associated with them is : (c being velocity of light) Answer: 1. 1/c(E/2m)^(1/2) Key idea: use λ=h/p for the electron and λ=hc/E for the photon. Electron: p=√(2mE) , so λ_e=h/√(2mE) . Photon: λ_... 1