Using the relativistic relationship between momentum and energy: a) Derive an expression for the wavelength of...
Using the relativistic relationship between momentum and energy:
a) Derive an expression for the wavelength of a particle with mass m in terms of its total energy.
b) Compare this result to the expression for the wavelength of a photon in terms of energy and show that as m → 0, the expressions are equivalent.
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Using the relativistic relationship between momentum and
energy:
a) Derive an expression for the wavelength of...
Derive the relationship for the momentum (p) of a relativistic mass bearing particle: p = (1/c)*[ET2...
Derive the relationship for the momentum (p) of a relativistic mass bearing particle: p = (1/c)*[ET2 – E02]0.5 and then, via De-Broglie’s hypothesis, derive the expression for the associated wavelength
Consider a relativistic particle of mass M and kinetic energy K. derive an expression for the...
Consider a relativistic particle of mass M and kinetic energy K. derive an expression for the particle's speed U in terms of K and M. show steps please
PHYS101 21 . a) Write down expressions for the total energy, E, and the momentum, p,...
PHYS101 21 . a) Write down expressions for the total energy, E, and the momentum, p, of a particle. Express your answers in terms of the rest mass, m, and velocity, u, of the particle. 4 marks) Using these expressions, show that E2-ap of the momentum. m2ct, where p Ipl is the magnitude (4 marks) b) A photon of frequency v is scattered trough 180° by an electron initially at rest in the laboratory. Show that the frequency of the...
B2 (a) Derive the Klein-Gordon equation (in S.I. units) starting from the energy-momentum relationship, E2 -mc4+kc2 usi...
B2 (a) Derive the Klein-Gordon equation (in S.I. units) starting from the energy-momentum relationship, E2 -mc4+kc2 using the quantum mechanical relations [3 Marks] (b) Write this in natural units [2 Marks] (c) Using the expression for the Laplacian in the radially symmetric case 8(3) r2 a show that the solution of the Klein-Gordon equation in the static case is (re-/R where R 1/m. You may wish to use the substitution [8 Marks] (d) Using the Heisenberg Uncertainty Principle, show that...
Ann measures the (relativistic) momentum of a tiny particle to be 3.2x 103 kg mIs. The...
Ann measures the (relativistic) momentum of a tiny particle to be 3.2x 103 kg mIs. The mass of the particle is 8 x 10-3g. and it is moving in the -x-direction. SP6 a. Find the total energy of the particle according to Ann b. Find the difference of the particle's total energy and its rest energy, and compare it to the kinetie energy of the particle calculated non-relativistically. Write the components of the momentum 4-vector of the particle according to...
с V A wave packet describes a particle having momentum p. Starting with the relativistic relationship...
с V A wave packet describes a particle having momentum p. Starting with the relativistic relationship E2 = p2c2 + Eo?, show that the group velocity is ßc and the phase velocity is where ß = ). How can the phase velocity physically be greater than c? The group velocity of a wave is represented by ugr dE which gives the following expression. (Use the following as necessary: C, P, and Eo.) Ugr dE dp The phase velocity of a...
1) Consider light in the near-infrared regime with a wavelength = 1000 nm. (a) What is...
1) Consider light in the near-infrared regime with a wavelength
= 1000 nm.
(a) What is the energy of a photon at this wavelength?
(b) Consider an electron that has a kinetic energy that equals the
photon energy in part (a). Calculate the electron momentum and
compare it to the momentum of the photon from part (a). Is the
momentum of the electron larger or smaller than the momentum of the
photon?
Hint: You can use non-relativistic equations for the...
(3) (10 pts): The work-energy theorem relates the change in kinetic energy of a particle to...
(3) (10 pts): The work-energy theorem relates the change in kinetic energy of a particle to the work done on it by an external force: AK = W = | Fdx. a) Writing Newton's second law as F=dp/dt, show that W = S v dp and integrate by parts using the relativistic momentum to obtain E = mc²y b) Use the expression for the relativistic energy and relativistic momentum of a particle of mass m to demonstrate the important relation...
Derive an expression for the spectral energy density ρλ(λ)[the energy per unit volume in the wavelength...
Derive an expression for the spectral energy density ρλ(λ)[the energy per unit volume in the wavelength region between λ and λ+dλ is ρλ(λ)dλ]. Show that the wavelength λp at which the spectral energy density is maximum satisfies the equation 5(1-e-y ) = y, where y=hc/λpkT, demonstrating that the relationship λpT = constant (Wien’s Law) is satisfied. Find λpT approximately. Show that λp ≠c/νp, where νp is the frequency at which the blackbody energy density ρv is maximum. The shapes and...
3. Derive the following relationship between the Helmholtz free energy F and the partition function Z for a system of N particles: (a) Starting with the thermodynamic definition F-U-TS, substitute th...
3. Derive the following relationship between the Helmholtz free energy F and the partition function Z for a system of N particles: (a) Starting with the thermodynamic definition F-U-TS, substitute the statistical mechanics results which give U and S in terms of occupation numbers n, state energies e and the most probable number of microstates t* to find, (b) Write out texplicitly in terms of occupation numbers using Stirling's approxima- tion (check the Lagrange multiplier derivation of the Boltzmann distribution)...
PHYS101 21 . a) Write down expressions for the total energy, E, and the momentum, p, of a particle. Express your answers in terms of the rest mass, m, and velocity, u, of the particle. 4 marks) Using these expressions, show that E2-ap of the momentum. m2ct, where p Ipl is the magnitude (4 marks) b) A photon of frequency v is scattered trough 180° by an electron initially at rest in the laboratory. Show that the frequency of the...
B2 (a) Derive the Klein-Gordon equation (in S.I. units) starting from the energy-momentum relationship, E2 -mc4+kc2 using the quantum mechanical relations [3 Marks] (b) Write this in natural units [2 Marks] (c) Using the expression for the Laplacian in the radially symmetric case 8(3) r2 a show that the solution of the Klein-Gordon equation in the static case is (re-/R where R 1/m. You may wish to use the substitution [8 Marks] (d) Using the Heisenberg Uncertainty Principle, show that...
Ann measures the (relativistic) momentum of a tiny particle to be 3.2x 103 kg mIs. The mass of the particle is 8 x 10-3g. and it is moving in the -x-direction. SP6 a. Find the total energy of the particle according to Ann b. Find the difference of the particle's total energy and its rest energy, and compare it to the kinetie energy of the particle calculated non-relativistically. Write the components of the momentum 4-vector of the particle according to...
с V A wave packet describes a particle having momentum p. Starting with the relativistic relationship E2 = p2c2 + Eo?, show that the group velocity is ßc and the phase velocity is where ß = ). How can the phase velocity physically be greater than c? The group velocity of a wave is represented by ugr dE which gives the following expression. (Use the following as necessary: C, P, and Eo.) Ugr dE dp The phase velocity of a...
1) Consider light in the near-infrared regime with a wavelength
= 1000 nm.
(a) What is the energy of a photon at this wavelength?
(b) Consider an electron that has a kinetic energy that equals the
photon energy in part (a). Calculate the electron momentum and
compare it to the momentum of the photon from part (a). Is the
momentum of the electron larger or smaller than the momentum of the
photon?
Hint: You can use non-relativistic equations for the...
(3) (10 pts): The work-energy theorem relates the change in kinetic energy of a particle to the work done on it by an external force: AK = W = | Fdx. a) Writing Newton's second law as F=dp/dt, show that W = S v dp and integrate by parts using the relativistic momentum to obtain E = mc²y b) Use the expression for the relativistic energy and relativistic momentum of a particle of mass m to demonstrate the important relation...
3. Derive the following relationship between the Helmholtz free energy F and the partition function Z for a system of N particles: (a) Starting with the thermodynamic definition F-U-TS, substitute the statistical mechanics results which give U and S in terms of occupation numbers n, state energies e and the most probable number of microstates t* to find, (b) Write out texplicitly in terms of occupation numbers using Stirling's approxima- tion (check the Lagrange multiplier derivation of the Boltzmann distribution)...
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