Differential equations

The solution of the differential equation for y(f): ix-y-2codtri) sulject 10 the initial conditions y(0) = 0 and dy =0. is dt liao (A) - cosh(l) +rsinh(t) (B) - sinh(f) + z cosh(f)...

Consider the linear differential equation $\frac{dy}{dx} = xy$. If $y=2$ at $x=0$, then the value of $y$ at $x=2$ is given by...

(A) If Yn(x) is a solution of the differential equation y" - 2xy' + 2ny = 0 where n is an integer and the prime (') denotes differentiation with respect to x, then acceptable plot(...

(A) (B) In a semiconductor, the ratio of the effective mass of hole to electron is 2:11 and the ratio of average relaxation time for hole to electron is 1:2. The ratio of the mobil...

Consider the ordinary differential equation $y'' - 2x y' + 4y = 0$ and its solution $y(x) = a + bx + cx^2$. Then...

The ordinary differential equation $(1 - x^2) y'' - x y' + 9y = 0$ has a regular singularity at...

Which one of the following is a dimensionless constant?...

For nonrelativistic electrons in a solid, different energy dispersion relations (with effective masses ma∗ , mb ∗ , and mc∗) are schematically shown in the plots. Which one of the ...

A 60Co nucleus emits a \beta-particle and is converted to 60Ni* with JP = 4+, which in turn decays to the 60Ni ground state with JP = 0+ by emitting two photons in succession, as s...

Which of the following options represent(s) linearly independent pair(s) of functions of a real variable x ?...

It is given that the electronic ground state of a diatomic molecule X2 has even parity and the nuclear spin of X is 0. Which one of the following is the CORRECT statement with rega...

A simple harmonic oscillator with an angular frequency \omega is in thermal equilibrium with a reservoir at absolute temperature T, with \omega = 2kBT \hbar . Which one of the foll...

A particle has wavefunction 𝜓(x, y, z) = N z e-\alpha(x2+y2+z2) , where N is a normalization constant and \alpha is a positi ve constant. In this state, which one of the following...

The wavefunction of a particle in one dimension is given by 𝜓(x) = {M, -a < x < a 0, otherwise. Here M and a are positive constants. If 𝜑(p) is the corresponding momentum space w...

Consider two non-identical spin 1 2 particles labelled 1 and 2 in the spin product state | 1 2 , 1 2〉 | 1 2 , - 1 2〉. The Hamiltonian of the system is H = 4λ \hbar2 S 1 ⋅ S 2 , whe...

Different spectral lines of the Balmer series (transitions n → 2, with n being the principal quantum number) fall one at a time on a Young's double slit apparatus. The separation b...

The wavefunction of a particle in an infinite one-dimensional potential well at time t is Ψ(x, t) = √2 3 e-iE1t \hbar⁄ 𝜓1(x) + 1 √6 ei𝜋 6⁄ e-iE2t \hbar⁄ 𝜓2(x) + 1 √6 ei𝜋 4⁄ e-i...

The symbols C, D, Vin and V0 shown in the figure denote capacitor, ideal diode, input voltage and output voltage, respectively. Which one of the following output waveforms (V0) is ...

A particle is subjected to a potential V(x) = { ∞, x ≤ 0 V0, a ≤ x ≤ b 0, elsewhere Here, a > 0 and b > a. If the energy of the particle E < V0, which one of the following schemati...

The temperature dependence of the electrical conductivity (𝜎) of three intrinsic semiconductors A, B and C is shown in figure. Let EA, EB and EC be the bandgaps of A, B and C, res...

Consider a volume integral I = ∫ ∇2 (1 r) dV V over a volume V, where r = √x2 + y2 + z2. Which of the following statement is/are correct?...

The Hamiltonian of a system of N particles in volume V at temperature T is H = ∑ aiqi 2 2N i=1 + ∑ bipi 2 2N i=1 where ai and bi are positive constants. The ensemble average of the...

The equation of motion for the forced simple harmonic oscillator is ẍ(t) + \omega2x(t) = F cos (\omegat) where x(t = 0) = 0 and ẋ(t = 0) = 0. Which one of the following options i...

An atom is subjected to a weak uniform magnetic field B⃗ . The number of lines in its Zeeman spectrum for transition from n = 2, l = 1 to n = 1, l = 0 is...

An infinite one dimensional lattice extends along x-axis. At each lattice site there exists an ion with spin 1 2 . The spin can point either in +z or - z direction only. Let SP, S ...

Consider a vector field F = (2xz + 3y2)ŷ + 4yz2ẑ. The closed path (Γ: A → B → C → D → A) in z = 0 plane is shown in figure. ∮ F . Γ dl⃗⃗⃗ denotes the line integral of F along the...

A particle of mass m is moving in the potential V(x) = {V0 + 1 2 m\omega0 2x2, x > 0, ∞ , x ≤ 0, Figures P, Q, R and S show different combinations of the values of \omega0 and V0. ...

The curves P and Q schematically show the variation of X -ray intensity with wavelength at two different accelerating voltages for a given target material. In the figure 𝜆1 = 0.25...

Apart from the acoustic modes, 9 optical modes are identified from the measurements of phonon dispersions of a solid with chemical formula AnBm, where A and B denote the atomic spe...

An oscillating electric dipole of moment d (t) = d0 cos(\omegat) ẑ is placed at origin as shown in figure. Consider a point P(r, 𝜃, \phi) at a very large distance from the dipole...

Consider a three-dimensional system of non-interacting bosons with zero chemical potential. The energy of the system \epsilon ∝ k2, where k is the wavevector. The low temperature s...

For a scalar field 𝜓(r ) and a vector field A (r ), ∇⃗⃗ × (A 𝜓) is equivalent to the expression...

The formula for energy E of a photon gas at temperature T in a two-dimensional box at equilibrium with g2d(𝜈) denoting the density of states of photons is given below where symbol...

For the electric field of an electromagnetic wave given below, which of the following statements is correct? E⃗ = x̂ E0cos(\omegat) + ŷ 2E0cos (\omegat + 𝜋 2)...

A gas of non-interacting 4He atoms (of mass m) is in a three-dimensional trap whose energy levels can be approximated by those of a harmonic oscillator potential V(x, y, z) = 1 2 m...

Given |v1⟩ = 1 √2 (1 i ) and |v2⟩ = 1 √2 ( 1 -i), the tensor product |v1⟩ ⊗ |v2⟩ is...

Sketch of a two-dimensional vector field V⃗ is shown below. Here, length and arrow head of the arrows denote magnitude and direction of the vector field, respectively. Which of the...

If the perturbation V = 𝜆x3 is added to the Hamiltonian of a one-dimensional harmonic oscillator, the matrix element ⟨m|V|0⟩ is/are non-zero for which of the following states? Her...

Consider the Pauli matrices 𝜎x = (0 1 1 0) , 𝜎y = (0 -i i 0 ) , 𝜎z = (1 0 0 -1). The value of Tr(𝜎z [ 𝜎x, 𝜎y]) is...

Which one of the following statements is true?...

The Hamiltonian for a quantum particle of mass m is given below, where \omega < Ω. The Schrödinger equation for this system can be solved exactly using the orthogonal transformati...

The energy E and degeneracy d of the second excited state of a three-dimensional, isotropic quantum harmonic oscillator with angular frequency \omega are...

Two identical particles with a fixed total energy E = 2\hbar\omega are in thermal equilibrium in a one-dimensional harmonic oscillator potential 1 2 m\omega2x2. Let the entropy of ...

An infinitely large thin sheet in the xy-plane carries uniform positive charge density and is moving with constant velocity v in the +x direction (see figure below). The direction ...

A capacitor is made of two circular metal plates of radius 1 m separated by a distance of d = 1 mm. The space between them is filled by a dielectric with permittivity \epsilonr = 5...

Copper has an electron number density of 8.3 × 1028 m-3. Its Fermi energy in eV (rounded off to one decimal place) is _____ (\hbar = 1.06 × 10-34 J. s, mass of electron me = 9.10 ×...