Quantum Physics
- What is de Broglie's hypothesis of matter waves and why was it proposed? In one line De Broglie proposed that every moving particle has a wave associated with it, with wavelength λ = h/p , just as light waves show... 1
- How is the de Broglie wavelength of an electron accelerated through a potential difference V derived? In one line An electron accelerated from rest through a potential difference V gains kinetic energy eV , so its de Broglie wavelength is λ =... 1
- What are the main properties of matter waves and how do they differ from light waves? In one line Matter waves are probability waves associated with moving particles; their wavelength is h/mv , they are not electromagnetic, an... 1
- What is a wave function and what does its probability density tell us about a particle? In one line The wave function psi is a mathematical function that describes the quantum state of a particle, and |psi|² gives the probabilit... 1
- What mathematical conditions must a wave function satisfy to be physically acceptable? In one line A physically acceptable (well-behaved) wave function must be single-valued, finite, continuous, have a continuous first derivati... 1
- How do you solve numerical problems on de Broglie wavelength for electrons and other particles? In one line Pick the right form of λ = h/p for the given data (speed, kinetic energy or accelerating voltage), substitute constants in SI un... 1
- Why was Schrödinger's equation needed and what is its significance in quantum mechanics? In one line Schrödinger's equation was needed because classical mechanics could not describe the wave behaviour of particles; it is the basi... 1
- How is Schrödinger's time-independent wave equation derived for a particle in one dimension? In one line Starting from the classical wave equation and substituting de Broglie's λ = h/p with p² = 2m(E - V) , we get d²psi/dx² + 8π²m/h²... 1
- How is the energy of a particle trapped in a one-dimensional rigid box derived using Schrödinger's equation? In one line Solving Schrödinger's equation for a particle confined between two infinitely high walls a distance L apart gives quantised ener... 1
- How is Schrödinger's time-dependent wave equation derived and what are the energy and momentum operators? In one line Differentiating the free-particle wave function Psi = Ae^(-i(Et - px)/hbar) gives the operators E → ihbar ∂/∂ t and p → -ihbar ∂... 1
- How do you solve numerical problems on the energy of a particle in a rigid box? In one line Use E_n = n²h²/(8mL²) in SI units for the required level, and use Δ E = E_(n_2) - E_(n_1) with λ = hc/Δ E for transitions betwee... 1
- What is quantum mechanical tunnelling and how can a particle cross a barrier higher than its energy? In one line Quantum tunnelling is the effect in which a particle has a non-zero probability of passing through a potential barrier even when... 1
- What is a qubit and how is it different from a classical bit in quantum computing? In one line A qubit is the basic unit of quantum information that can exist in a superposition of the states |0rangle and |1rangle at the sa... 1
- An electron has a speed of 400 m/sec with uncertainty of 0.01%. Find the accuracy in its position. Given: v=400 m/s, so Δ v=0.01% of 400 =0.01/100×400=0.04 m/s. m=9.1×10⁻³¹ kg. Uncertainty principle: Δ x Δ p≥h/4π Δ x=h/4π mΔ v=frac6.63×10⁻... 1
- Calculate the wavelength of de-Broglie waves associated with mass 1 kg moving with a speed of 103 m/sec. Given: m=1 kg, v=10³ m/s, h=6.63×10⁻³⁴ J s. Formula: λ=h/mv λ=frac6.63×10⁻³⁴1×10³=6.63×10⁻³⁷\ m λ=6.63×10⁻³⁷ m This is far too small to dete... 1
- Explain the de Broglie hypothesis of mater waves. de Broglie hypothesis (1924): Radiation has a dual nature: waves (interference) and particles (photoelectric effect). Nature loves symmetry,... 1
- Derive Schrodinger’s Time Independent wave equation for matter waves. The wave associated with a particle moving along x can be written as psi(x,t)=A e^(i(kx-ω t))=psi₀(x) e^(-iω t) where psi₀(x)=Ae^ikx . Diffe... 1
- The ground state energy of an electron in an infinite well is 1 eV. What will be the ground state energy if the width of the well is doubled? Formula: E_n=n²h²/8mL² , so for the ground state E₁∝1/L² . E₁'/E₁=Big(L/L'Big)²=Big(L/2LBig)²=1/4 E₁'=1/4×1\ eV New ground-state energy = 0.... 1
- Write the properties of matter waves. Properties of matter waves (de Broglie waves): Wavelength λ=h/p=h/mv : lighter particles have larger wavelengths, and so do slower ones. If... 1
- Derive the Schrodinger’s Time Dependent differential equation for matter waves A free particle moving along x is described by the wave function psi(x,t)=A e^(-i(Et-px)/hbar) using E=hbarω and p=hbar k . Step 1 (time der... 1
- Calculate velocity and de-Broglie wavelength of alpha particle of energy 1[unit prefix partly illegible in the source scan; read as k]eV. Given: mass of alpha particle as 6.68x10-27 kg. Assumption: The energy unit is partly obscured in the scan. We read it as E=1 keV. The results for 1 MeV are given at the end. Given: E=1 ke... 1
- State the properties of matter waves. Properties of matter waves (de Broglie waves): Wavelength λ=h/p=h/mv : lighter particles have larger wavelengths, and so do slower ones. If... 1
- An electron is bound in an 1D potential well of width 2 A° but of infinite height. Find its energy values in the ground state and in first excited state. (Given: Planck constant is 6.634x10-34 J-Sec) Given: L=2 Å =2×10⁻¹⁰ m, h=6.634×10⁻³⁴ J s, m=9.1×10⁻³¹ kg. Formula: E_n=n²h²/8mL² Ground state (n = 1): E₁=frac(6.634×10⁻³⁴)²8×9.1×10⁻³¹×(2... 1
- Prove that electron cannot reside inside the nucleus of an atom. Using Heisenberg's uncertainty principle: Δ x Δ p≥hbar/2=h/4π The diameter of a nucleus is about 10⁻¹⁴ m. If an electron were confined insid... 1