Quantum Mechanics

Step 1: Write down the general form of the time-dependent Schrodinger equation. where is the wave function.
Step 2: Analyze the properties of the equation. - **Linearity:** The equation is linear because the wave function and its derivatives appear only to the first power. There are no terms like or . This means that if and are solutions, then any linear combination is also a solution. This is the principle of superposition. - **Order in time:** The equation contains a first derivative with respect to time, . It is first-order in time. - **Order in space:** The equation contains the Laplacian operator, , which involves second derivatives with respect to spatial coordinates. It is second-order in space.
Step 3: Evaluate the given options. 1. non-linear differential equation: Incorrect. 2. linear differential equation: Correct. 3. second order equation in time: Incorrect. It is first-order in time. 4. first order equation in space: Incorrect. It is second-order in space.

Step 1: Recall the de-Broglie wavelength formula. The de-Broglie wavelength ( ) of a particle is given by: where is Planck's constant, is the mass of the particle, and is its velocity.
Step 2: List the required constants and given values. - Planck's constant, . - Mass of an electron, . - Velocity of the electron, .
Step 3: Substitute the values and calculate the wavelength. This value is approximately m.

Step 1: Recall the energy formula for a particle in a 3D cubic box. The energy levels for a particle of mass in a cubic box of side length are given by: where are positive integers (1, 2, 3, ...). Degeneracy is the number of different combinations of that yield the same energy value.
Step 2: Calculate the degeneracy for each energy level. - A. Energy : We need to find the number of integer sets such that . By inspection, . The distinct sets are permutations of (1, 2, 3). The number of permutations of three distinct numbers is . The states are (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1). The degeneracy is 6. So, A III. - B. Energy : We need . By inspection, . The distinct sets are permutations of (1, 1, 3). The number of permutations is . The states are (1,1,3), (1,3,1), (3,1,1). The degeneracy is 3. So, B II. - C. Energy : We need . The only possible combination is . The state is (1,1,1). There is only one such state. The degeneracy is 1. So, C I.
Step 3: Formulate the correct matching sequence. The correct matches are: A III, B II, and C I. This corresponds to option (3).

In quantum mechanics, particles are classified into two fundamental types based on their intrinsic angular momentum, or spin. - Fermions: These are particles that have half-integer spin values ( ) in units of . They obey the Pauli exclusion principle, which states that no two identical fermions can occupy the same quantum state simultaneously. Examples include electrons, protons, and neutrons (all have spin 1/2). - Bosons: These are particles that have integer spin values ( ) in units of . They do not obey the Pauli exclusion principle. Examples include photons (spin 1) and Higgs bosons (spin 0). The option provided that is a half-integer spin is .