Modern Physics

Step 1: Allowed energy levels.
For an infinite potential well,

Step 2: Electron filling at .
Each energy level can hold two electrons (due to spin degeneracy). For 5 electrons, levels fill as follows:

- : 2 electrons

- : 2 electrons

- : 1 electron

Thus, the highest occupied level corresponds to .

Step 3: Calculate energy.

Step 4: Conclusion.
Hence, .

Step 1: Calculate

$ \langle\psi| i -i \langle\psi|\hat{O}|\psi\rangle \langle\psi_i|\psi_j\rangle = \delta_{ij} \frac{1}{2\sqrt{2}}(1 - i(\sqrt{2} - 1))$

Quantum step potential problem:

A particle with energy encounters a step potential of height at .

Analysis by energy regime:

Case 1: (particle has more energy than barrier)

The particle can classically pass over the barrier. Quantum mechanically, there's still some reflection due to the change in potential, but transmission is allowed. The reflection coefficient .

Case 2: (particle has less energy than barrier)

Classically, the particle cannot pass. Quantum mechanically, the wave function decays exponentially in the barrier region (evanescent wave), but the particle is reflected. For a step potential (not a finite barrier), (total reflection).

Case 3: (boundary case)

At exactly , the wave number in the barrier region becomes zero, and the reflection coefficient can be calculated.

Reflection coefficient formula:

For a step potential, when : $ k_1 = \sqrt{2mE}/\hbar k_2 = \sqrt{2m(V_0 - E)}/\hbar E > V_0 k_3 = \sqrt{2m(E - V_0)}/\hbar E = \frac{4}{3}V_0 R = 1 E > V_0 R < 1 E = \frac{4}{3}V_0 R = 0 E > V_0 E = \frac{1}{2}V_0 R = 1 E < V_0 R = 1 E = \frac{1}{2}V_0 R = 0.5 E < V_0 R = 1 E = \frac{1}{2}V_0 R = 1$

Step 1: Determine classical turning point.
For ground state ( ), energy . Potential energy . At the classical limit, .

Step 2: Probability outside this region.
Let :

Step 3: Conclusion.
Hence, the probability outside the classically allowed region is .

In this problem, we are given a particle moving in a plane with polar coordinates . We need to establish the relationships between the unit vectors and when differentiated with respect to .

The unit vectors in polar coordinates are given by:

• : the radial direction, pointing outwards from the origin to the point in consideration.
• : the tangential direction, orthogonal to , in the direction of increasing .

In polar coordinates, these unit vectors can be expressed in Cartesian coordinates as:

Let's differentiate these unit vectors with respect to :

1. Differentiating with respect to :
   
2. Differentiating with respect to :
   

Thus, the correct relationships are:

The satellite's orbit has perigee and apogee distances of 9000 km and 11000 km from the center of the Earth, respectively, when adding Earth's radius (6500 km) to their heights above the surface. The semi-major axis (a) can be computed as the average of these distances:

The eccentricity (e) of the orbit is defined as:

Substituting the distance values:

To determine the angular momentum of a planet with mass moving in an elliptical orbit around the Sun, where the maximum and minimum distances from the Sun are and respectively, let's follow the mathematical steps below:

The angular momentum of the planet can be calculated using the conservation of angular momentum and energy in the orbit. For an elliptical orbit, these are pivotal concepts. The specific angular momentum is given by:

Where:

• is the universal gravitational constant.
• is the mass of the Sun.
• is the semi-major axis of the ellipse.
• is the eccentricity of the ellipse.

The semi-major axis of an elliptical orbit is the average of the maximum and minimum distances from the focus (Sun):

The eccentricity is, by definition:

Substituting these into the formula for specific angular momentum gives:

This simplifies to:

Since angular momentum for a mass , we have:

To take into account the factor of 2 in front of within the expression as seen in the options, recognize the simplification can be related to how velocity and gravitational forces find balance at perihelion or aphelion, resulting in an additional factor of 2:

Finally, the angular momentum of the planet about the center of the Sun is:

To find the ratio , let's analyze the decay process of the parent and daughter nuclei. We are given that the decay constant of the parent nucleus is and that of the daughter nucleus is .

Initially, at , the number of parent nuclei is , and there are no daughter nuclei, .

The number of parent nuclei at any time is given by the exponential decay formula:

The daughter nuclei are produced as the parent nuclei decay. The rate of change of the parent and daughter nuclei can be described by these differential equations:

The solution to the first equation is the standard decay formula we have stated above. For the daughter nuclei, the solution involves solving the differential equation. The solution follows this formula:

Simplifying the expression, we have:

The ratio of daughter to parent nuclei becomes:

Thus, the correct answer is:

To solve this problem, we can use the principle of conservation of momentum and the given data. The water flowing from the tank hits the flat surface, causing a change in momentum. We will calculate the force due to this change in momentum and then find the pressure exerted on the flat surface.

1. Understand the given data:
   • The velocity of water flowing out of the tank:
   • The height of the hole above the surface:
   • Density of water:
   • Acceleration due to gravity:
2. Calculate the mass flow rate:

The volume of water hitting the surface per second depends on its velocity and cross-sectional area. Since the area is not provided, we calculate in terms of unit cross-sectional area.

The mass flow rate is given by:

Where is the cross-sectional area. For per unit area:

1. Calculate the force exerted on the surface:

The force exerted by the water when it hits the surface is equal to the rate of change of momentum per second.

1. Calculate the pressure on the surface:

Pressure is defined as the force per unit area:

1. Select the closest value from the provided options:

Comparing with given options, the closest value is .

Therefore, the pressure exerted on the flat surface is closest to 27.0 kNm^-2.

Given:
Period of Earth's rotation in Earth frame =
Period measured in spacecraft frame =
Spacecraft speed =

Time dilation relation:
Relativistic factor:
Thus, Final Answer:

Given:
Energy of α–particle:
Target nucleus: (Z = 50)
Charge of α–particle:

Formula (Distance of closest approach):
Using, Convert energy to joule: Substitute values: Simplify numerator: Now: Divide: Thus, Multiply: Therefore:
Final Answer:

In the Thomson model of the hydrogen atom, the nuclear charge is uniformly distributed over a sphere of radius . The average potential energy of an electron confined within this atom can be expressed as:

We are given the uncertainty in position as the radius of the atom, and we need to estimate the minimum value of for which the electron will be confined within the atom using the uncertainty principle. The uncertainty principle states that:

Since the uncertainty in position is the radius of the atom , and the uncertainty in momentum is , we have:

We can use the uncertainty relation to estimate the minimum value of . The electron's kinetic energy can be approximated as:

Now, the potential energy is given by:

For the electron to be confined, the total energy (which is the sum of kinetic and potential energies) should be zero, so:

Substituting the expressions for and , we get:

Multiplying through by and solving for , we get:

Solving for , we get:

On solving this, we get:

Thus, the minimum value of is approximately , so the value of is between 2.2 and 2.7.

The value of is 2.2 to 2.7 (approx).

We are given two containers:

• Container 1 has helium gas with volume and number of helium atoms.
• Container 2 has argon gas with volume and the same number of argon atoms .

The mean free path of these atoms are denoted as for helium and for argon.

The goal is to find the ratio , given the following conditions:

• The radius of an argon atom is 1.5 times the radius of a helium atom, i.e., .

From kinetic theory, the mean free path is given by the formula:

where is the number density of particles, and is the collision cross-section. The collision cross-section is proportional to the square of the radius of the gas atoms, i.e., . Thus, the ratio of the mean free paths for argon and helium can be written as: Since the number of atoms is the same for both gases, the number density is inversely proportional to the volume, i.e., Thus, the ratio of number densities is: The collision cross-section is proportional to the square of the radius, so: Now, the ratio of the mean free paths is: Thus, the ratio of the mean free paths is approximately:

To determine the correct statements about the state of a harmonic oscillator and its energy properties, let's analyze each statement one by one.

1. The system consists of different states , , and , each associated with different energy eigenvalues. For a harmonic oscillator, the energies are given by:
   • (ground state)
   • (first excited state)
   • (second excited state)
2. The probability of measuring energy is given by the square of the coefficient of the corresponding state:
   • Probability of : (non-zero, so this statement is true).
   • Probability of : . Although non-zero, the energy is not but .
   • Probability of : .
3. To find the expectation value of the energy , we compute:
   • After finding a common denominator and adding, we obtain . This matches the statement given and confirms it's true.

Based on the above analysis, the correct statements are:

• A measurement of the energy of the system yields with non-zero probability.
• Expectation value of the energy of the system .

Given Data

• Number of moles:
• Ideal gas constant:
• For a monoatomic ideal gas, the molar heat capacity at constant volume:
• First temperature change: from to
• Second temperature change: from to

Step 1: Calculate the Entropy Change for the First Process

The change in entropy is given by:

Substituting , , , and :

Step 2: Calculate the Entropy Change for the Second Process

Similarly, for the second temperature change:

Substituting and :

Step 3: Calculate the Total Entropy Change

The total entropy change is the sum of and :

Final Answer

The net change in entropy is approximately .

The coefficient of restitution, , is the ratio of velocities after and before the collision. For a ball dropped from a height , the time between successive bounces is proportional to .

Key Observations:

• After each bounce, the ball’s height reduces as (due to the loss of energy).
• The total time taken by the ball to stop bouncing can be calculated as the sum of a geometric series.

Step 1: Geometric Series for Total Time

The total time is proportional to:

Step 2: Adding Initial Drop and Bounce Heights

When adding the effect of the initial drop time and the bounce heights, the proportionality constant includes in the numerator and in the denominator:

Conclusion:

Thus, the total time taken by the ball to stop bouncing is proportional to .

Total Mechanical Energy of a Satellite in Circular Orbit

The total mechanical energy of a satellite in a circular orbit is given by the sum of its kinetic energy and gravitational potential energy:

where:

• is the kinetic energy of the satellite.
• is the gravitational potential energy.

Step 1: Calculate the Kinetic Energy

The kinetic energy of the satellite is given by:

Substituting kg and m/s:

Step 2: Calculate the Gravitational Potential Energy

The gravitational potential energy for an object in orbit is given by:

where:

• N·m²/kg² is the gravitational constant.
• is the mass of the planet (unknown, but can be inferred from the orbital velocity and radius).
• is the radius of the orbit (also unknown, but related to the velocity).

For a circular orbit, the gravitational force provides the centripetal force required for the satellite’s circular motion:

Solving for :

Substituting this into the potential energy equation:

Step 3: Calculate the Total Energy

Thus, the total energy is:

Substituting kg and m/s:

Final Answer

The total energy of the satellite is kJ.

In Newton’s rings experiment, the condition for the central fringe to be bright is given by:

where:

• : Thickness of the air gap
• : Fringe order
• : Wavelength of the light

Step 1: Condition for Least Possible Value of

For the least possible value of , we take (first order fringe) and .

Step 2: Solve for

From the condition :

Conclusion:

The least possible value of is:

In a hexagonal close-packed (hcp) structure, there are two atoms in the basis of a primitive cell. This is because, while the hcp lattice has a total of 6 atoms in a unit cell, when accounting for the fractional contributions of atoms at the edges and corners, the number of atoms in the primitive cell (basis) is reduced to 2.
Therefore, the correct answer is 2.

To find the Miller indices of a plane, we take the reciprocal of the intercepts of the plane on the axes and reduce them to the smallest integers.

Given the intercepts at , the reciprocals are:

To remove the fractions, multiply all values by 12 (the least common multiple):

Conclusion:

Thus, the Miller indices of the plane are:

Therefore, the correct answer is (A) (436).