How can the Heisenberg Uncertainty Principle be mathematically expressed?
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ΔE . Δt = ħ / 2π.
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How can the Heisenberg Uncertainty Principle be mathematically expressed?
ΔE . Δt = ħ / 2π.
What does the wave nature of matter introduce in terms of particle location?
An uncertainty in the location of the particle.
Who provided the first experimental evidence of matter waves?
Davisson and Germer in 1927.
What is the famous formula for Black body radiation?
E = (8πhc/λ^5)(e^(hν/kT) - 1)
What happens to the bump in the curve with increasing potential?
The bump moves upwards.
What is the normalized wave function ψ(x) after substituting A?
ψ(x) = √(2/L) sin(2nπx/L).
How is kinetic energy expressed in terms of momentum?
2mKE = p² (where p = mv)
What is the result of integrating sin²(2nπx/L) over the interval [0, L]?
L/2.
What does the wave function ψ represent?
It is a complex quantity representing the variation of matter wave.
What is the equation used to calculate the unknown constant A?
A² ∫ sin²(2nπx/L) dx = 1.
What is the value of the constant A after normalization?
A = √(2/L).
What are valence electrons?
Electrons in the outermost shell that are free to move.
What theory did Huygens propose in 1979 to explain light phenomena?
The wave theory of light.
What is the relationship between the total emissive power (R) and absolute temperature (T) according to Stefan's Law?
R(T) = σT^4, where σ is Stefan’s Constant (5.67 x 10^-8 wm^-2 k^-4).
What occurs to the bumps at higher potentials?
The bumps gradually disappear.
What does the most prominent bump verify?
The existence of electron waves.
What is the physical significance of |ψ|²?
It is the probability density function.
What does the Kronig-Penney model suggest about the potential energy of electrons in a crystal?
It varies due to the presence of immobile lattice ions.
At what potential does the bump become most prominent?
54 volts.
What principle states that measuring the position accurately affects the measurement of momentum?
The Heisenberg Uncertainty Principle.
What are core electrons?
Electrons in the closed shell that are strongly attracted by the nucleus.
What is the purpose of the electron gun in Davisson-Germer's experiment?
To produce electrons through thermionic emissions.
What happens to the electrons after they are emitted from the electron gun?
They are accelerated in an electric field of known potential difference.
What does a nickel crystal act as in X-ray analysis?
A plane diffraction grating.
What is the term for the values corresponding to each energy value En?
Eigen values.
What is the product of the uncertainties in position and momentum according to the treatment provided?
Δx . Δp = ħ.
What is the relationship between kinetic energy and potential difference for a charged particle?
eV = 1/2 mv²
What does the classical wave equation in differential form describe?
The behavior of a particle with wave properties.
What is the wavelength associated with an electron accelerated through 54 volts?
1.67 Å.
What is defined as the spreading of disturbance in a medium?
Wave.
What does Wien's Law state about the wavelength corresponding to maximum energy (λm)?
λm is inversely proportional to absolute temperature (T), i.e., λmT = constant.
What is the boundary condition applied to the wave function in a box?
At the boundary, ψ = A sin(Kx) must equal zero.
What is drift velocity?
The average velocity acquired by free electrons in a particular direction after an electric field is applied.
What does ΔE represent in the Heisenberg Uncertainty Principle?
The error in the measurements of energy.
How is kinetic energy related to temperature?
E = 3/2 kT (where K is the Boltzmann constant)
What is the expression for momentum in terms of temperature?
p = √(3mKT)
What does ψ₀ represent in the wave function?
The amplitude at the point considered.
What is the De Broglie wavelength in terms of kinetic energy?
λ = h / √(2mKE)
What does the electrical conductivity of a metal measure?
The amount of electrical charge conducted per unit time across unit area per unit applied electrical field.
What does the integral ∫ψψ* dx dy dz equal if the particle is present?
1, known as the normalized condition of wave function.
In the Kronig-Penney model, where is the potential of an electron at the positive ion site?
Zero, and it is maximum between two ions.
What does Δx represent in the context of the microscope?
The uncertainty in the measurements of the position of the electron.
How is the angle of the target adjusted in the experiment?
The target is rotated about an angle along the direction of the beam.
What is the significance of substituting ω = 2πν in the wave equation?
It relates angular frequency to frequency.
What is a drawback of classical free electron theory regarding energy absorption?
It states that all free electrons absorb the supplied energy, while quantum theory states that only a few do.
What is the formula for the de-Broglie wavelength of an electron?
𝜆 = h / √(2mE), where h = 6.626 x 10^-34 Js, m = 9.1 x 10^-31 Kg, and e = 1.6 x 10^-19 C.
What is the calculated de-Broglie wavelength of an electron?
𝜆 = 12.26 x 10^-10 m.
What is the form of the Schrödinger's equation for the region where 0 < x < a?
0 = (d²ψ/dx²) + (2m/ħ²)Eψ.
What is the role of free electrons in the classical free electron theory?
They are responsible for electrical conduction.
What is the normalization condition for the wave function ψ(x)?
∫ ψ(x)² dx = 1.
What is the equation representing the periodic potential in the context of quantum mechanics?
V(x) = V(x + a), where 'a' is the lattice constant.
What is the De Broglie wavelength in terms of potential difference?
λ = h / √(2meV)
What is the formula for electrical conductivity?
σ = Q / (t * A * E) = J / E
What is Fermi energy?
The energy of the state at which the probability of electron occupation is ½ at any temperature above 0K; it is the maximum energy of filled states at 0K.
What is relaxation time (τ)?
The average time taken by a free electron to reach its equilibrium position from a disturbed position due to an external electric field, approximately 10^-14 seconds.
What are some phenomena that classical theories could not explain?
Compton Effect, Photoelectric Effect, Zeeman Effect, black body radiation.
What is the relationship between the change in momentum of an electron and the scattering of a photon?
Δp = 2ħ sinθ.
What is the value of the constant in Wien's Law?
2.98 x 10^-3 mK.
What is the form of the wave function inside the box?
ψ(x) = A sin(Kx) + B cos(Kx).
What boundary condition applies when x = 0?
The wave function ψ = 0.
What is the Rayleigh-Jeans spectral distribution formula?
E(λ) = (8πKT/λ^4).
What does the infinite potential energy outside the box imply?
The particle cannot escape from the box.
What is the significance of the wave function normalization?
It ensures that the total probability of finding the particle is equal to one.
How does the frequency of radiation relate to the vibrating particles in Planck's theory?
The frequency of radiation is the same as that of the vibrating particles.
What is the form of the solution to Schrödinger's equation proposed by Bloch?
Ψₖ(x) = e^(±ikx) Uₖ(x), known as the Bloch function.
What does Uₖ(x) represent in the context of Bloch functions?
A periodic function with the periodicity of the crystal lattice.
What was the prevailing belief about physics until the end of the nineteenth century?
Classical physics was considered sufficient for all physical phenomena.
What does Δt represent in the Heisenberg Uncertainty Principle?
The error in the measurements of time.
What is the formula for the resolving power of a microscope in measuring position?
Δx = λ / (2sinθ).
What is the target used in the Davisson-Germer experiment?
A large single crystal of nickel.
How did Kronig and Penny modify the potential energy curves for studying electron behavior?
They represented them as rectangular square potential wells.
What is the potential energy at the walls of the box?
Infinite.
What is mean free path?
The average distance traveled by a free electron between two successive collisions.
What observation is made about the energy distribution in the spectrum of a black body?
The distribution of energy is not uniform.
How can a perfect black body be approximated?
By using a cavity with blackened walls, where a small hole behaves like a black body surface.
What does Kirchhoff's law imply about the emissive power of a black body?
It does not depend on the nature of the body, indicating universal properties.
What is the total energy of a photon considered as a wave?
E = hν.
What are the values of energies in Planck's hypothesis?
0, hν, 2hν, 3hν, ..., nhν.
What is the significance of the oscillators moving from one state to another in Planck's theory?
They can radiate energy during this transition.
What is the equation known as Schrodinger’s time-independent wave equation for one-dimensional motion?
𝜕²ψ/𝜕x² + (2m/ℏ²)(E - V)ψ = 0.
What is the de-Broglie wavelength formula for a particle of mass m moving with velocity v?
λ = h/mv.
According to the free electron theory, how is the potential of an electron inside a crystal described?
It is uniform.
What are the characteristic parameters of a particle?
Mass, velocity, momentum, and energy.
What are the discrete values of energy for a particle in a box called?
Quantized energies.
What does Bloch's theorem describe?
It describes how conduction electrons move throughout a crystalline lattice composed of ionic cores.
What is the relationship between total energy E, potential energy V, and kinetic energy in the context of the equations?
E = V + (½ mv²) or E - V = (½ mv²).
What is the corresponding wave function for each energy value called?
Eigen function.
What does the Heisenberg Uncertainty Principle state?
The product of uncertainties in position and momentum is equal to ℏ/2π.
What is the energy of a photon given by?
E = hν, where ν is the frequency of radiation.
What is collision time?
The average time taken by a free electron between two successive collisions.
What happens to the intensity of radiation for a particular temperature as wavelength increases?
It increases up to a particular wavelength and then decreases.
What occurs to the peak energy as temperature increases?
The peak energy shifts towards shorter wavelengths.
How is the wave velocity 'u' of matter waves obtained?
From the photon energy as u = frequency x wavelength.
What is the relationship between particle velocity and de-Broglie wavelength?
Lesser the velocity of the particle, longer the wavelength associated with it.
What is the speed relationship of matter waves compared to the speed of light?
Matter waves travel faster than the velocity of light, but the particle velocity cannot exceed the speed of light.
What is the formula for the wavelength associated with an electron accelerated through a potential V?
λ = 12.26 √V Å.
What is the general form of the wave function ψ(x, t) according to the solution of the wave equation?
ψ(x, t) = ψ₀(x)e^{-iωt}.
How is the wave function ψ used in quantum mechanics?
It is considered as probability amplitude to find the location of the particle.
What happens to the wave function outside the box?
The wave function is zero (|ψ|² = 0 for 0 > x > L).
What are the demerits of quantum free electron theory?
It fails to distinguish between metals, semiconductors, and insulators, and does not explain the positive value of Hall coefficient and some transport properties of metals.
According to quantum theory, how is radiation emitted?
In the form of photons.
What equation describes the motion of the electron in the one-dimensional box?
The Schrödinger wave equation.
What did Rayleigh and Jeans suggest about electromagnetic radiation?
It is caused by the constant absorption and emission of radiation by atoms in the wall of the cavity.
What is the potential energy (PE) of an electron in the well?
V = 0 in the well.
What is the potential energy (PE) of an electron in the barrier?
V = Vo in the barrier.
What concepts explain the energy distribution in black body radiation?
Stefan’s fourth power law, Wien’s law, and Rayleigh-Jeans law.
What is a key assumption of quantum free electron theory regarding electron movement?
Electrons move in a constant potential within the crystal.
What surrounds the positively charged nucleus in an atom?
Negatively charged electrons.
What did Drude assume about electrons in a metal?
That they are free to move and form an electron gas.
What is the potential energy of the particle inside the one-dimensional box?
Zero (V = 0 for 0 < x < L).
What happens to the wave function ψ when x = L?
ψ = 0, which implies ψ² = 0.
What is Bragg's equation used for in this context?
To calculate the wavelength from diffraction patterns.
What does the equation ∂²ψ/∂t² = -ω²ψ represent?
The relationship between time and wave displacement in wave mechanics.
What law does classical free electron theory help to derive?
Weidman-Franz law.
What does ℏ represent in the equations?
ℏ = h/2π, where h is Planck's constant.
What type of collision occurs among free electrons?
Elastic collision.
What happens to electrons when an external electric field is applied?
They acquire energy and move towards the positive potential, resulting in drift velocity.
Who proposed the quantum free electron theory and when?
Somerfield proposed it in 1928.
What statistical method does quantum free electron theory use?
Fermi-Dirac statistics.
What does the equation abmVo = P represent?
It relates the potential energy of the barrier to the wave function parameters.
What does the figure mentioned illustrate?
The energy level diagram for the particle.
What is the concentration of free electrons in a metal?
10^28 m^-3.
What is the formula for De Broglie wavelength in terms of temperature?
λ = h / √(3mKT)
What is the result of differentiating the wave function with respect to time?
∂²ψ/∂t² = -ω²ψ₀(x)e^{-iωt}.
What is black body radiation considered in quantum physics?
The first theory of quantum physics.
What device is used to detect scattered electrons in the experiment?
A Faraday cylinder connected to a galvanometer.
What are the two physical quantities described by the Heisenberg Uncertainty Principle?
Position (Δx) and momentum (Δp).
What concept did de Broglie suggest in 1924?
That particles like electrons exhibit wave-like properties.
What are waves associated with material particles called?
Matter Waves.
What is the total energy of a photon considered as a particle?
E = mc².
What are the integer values n known as?
Quantum numbers of energy levels Eₙ.
What does the term 'K' represent in the wave function?
K is the propagation vector, K = 2π/λ, where λ is the de Broglie wavelength of the electron.
Who introduced the electromagnetic theory?
Maxwell.
What is one merit of classical free electron theory?
It is used to verify Ohm’s law.
How is the potential energy of a conduction electron in a lattice characterized?
It is minimum at the positive ion sites and maximum between the two ions.
What is the expression for the energy eigenvalues E?
E = (n²ħ²π²)/(2mL²).
What is the significance of the 'bump' observed in the galvanometer current?
It indicates the diffraction pattern of electrons, supporting the wave nature of matter.
What is the relationship between wave number k and wavelength λ?
k = 2π/λ.
What does the spacing between energy levels depend on?
The spacing increases as (2n + 1)E₁.
What is the nature of energy radiation according to Planck's theory?
The radiation of energy is discrete, not continuous.
What does it mean when v = 0 in terms of de-Broglie wavelength?
When v = 0, λ = ∞, indicating that matter waves are generated by the motion of particles.
What happens to valence electrons when atoms are brought closer to form a metal?
They get detached and move freely through the metal.
What happens to λm when the temperature of the black body increases?
λm shifts towards the minimum value.
How does classical free electron theory explain the conductivity of metals?
It explains the electrical and thermal conductivity of metals.
What did G.R. Kirchhoff prove in 1959 regarding black body radiation?
The ratio of emissive power to absorption coefficient is the same for all bodies at the same temperature.
What is a black body defined as?
A body that absorbs all radiant energy falling upon it and emits all wavelengths of radiation when heated.
What are the three types of conducting materials based on conductivity?
Zero resistivity, low resistivity, and high resistivity materials.
What does the Electron Theory of Solids explain?
The structure and properties of solids through their electronic structure.
What did Planck suggest in 1900 to explain the energy distribution in black body radiation?
He proposed a new hypothesis regarding discrete energy oscillations of electrons.
What does the quantum free electron theory obey?
The laws of quantum mechanics.
What is the relationship derived from differentiating the wave function and substituting into the Schrödinger's equations?
ka = αa + αa * cos(2mVo/ħ²) * sin(αa).
What are the characteristics of waves?
Amplitude, time period, frequency, wavelength, phase, and intensity.
What are low resistive materials known as?
Conducting materials.
What determines the conducting property of a solid?
The number of valence electrons.
What is the average energy (ε) of oscillators in thermal equilibrium according to classical statistical mechanics?
ε = KT/2, where K is Boltzmann’s constant.
What is the relationship between K and the mass of the particle?
K² = 2mE/ħ².
What is the classical free electron theory based on?
The movement of electrons in a lattice obeying classical mechanics.
What is the form of the Schrödinger's equation for the region where -b < x < 0?
0 = (d²ψ/dx²) - (2m/ħ²)(Vo - E)ψ.
Are matter waves electromagnetic waves?
No, matter waves are not electromagnetic waves; they are pilot waves guiding the particle.
What does the equation KL = nπ represent?
It represents the quantization condition for the wave function in a box.
What is observed in the galvanometer current as the angle is varied?
A 'bump' begins to appear in the curve for certain acceleration potentials.
What does the equation ∂²ψ/∂x² = -k²ψ represent?
The wave equation in terms of spatial displacement.
What phenomenon cannot be explained by classical free electron theory?
The photoelectric effect, Compton effect, and blackbody radiation.
What is the experimental value of specific heat of a metal compared to classical theory?
Classical theory predicts 4.5R, while the experimental value is 3R.
Who developed the Zone Theory (Band Theory)?
Bloch.
What assumption did Planck make about the particles in a black body?
They are oscillating particles that can vibrate at all possible frequencies.
What are the widths of the potential well and potential barrier in the model?
The width of the potential well is 'a' and the potential barrier is 'b'.
What is the lowest energy of the particle in the box?
E₁ = (ħ²π²)/(2mL²), known as zero point energy.
How are the energy levels of the particle in the box quantized?
The energy levels are discrete and given by Eₙ = n²E₁.
What equations are used to calculate the energies and wave functions of the electron?
The time-independent one-dimensional Schrödinger’s wave equations.
What happens to the de-Broglie wavelength as the mass of the particle decreases?
The lighter the particle, the greater the wavelength associated with it.
What does the equation (x) u e = ψ(x) represent?
It represents the wave function of the electron in the potential well and barrier.
