Question 2 In class, we defined as the effective resolution for counting positions of non-interacting particles moving in three dimensions. Why is this the minimum resolution that we consider? O λ is defined as the size of the particle. We cannot have microstates smaller than the size of the particle. O A is a made up resolution that helps with the math. O For an ideal gas, if particles get too close together, we need to start considering particle-particle interactions. 1 pts O The Heisenberg uncertainty principle dictates our minimum resolution because it is difficult to precisely know the velocity and position of particles below a certain limit.

4. A one-particle, one-dimensional system has the state function. 1 e y = sin(at) (2) 4: 32 1 + cos (at) (2) 4 x xe Where a is a constant and c-2.000 Å. If the particle's position is measured at t=0, estimate the probability that the result will between 2.000 Å and 2.0001 Å. (This range can be thought of as infinitesimal process)

2. I don't exactly understand what this question is trying to do or ask or how to even solve it. What does the Gcalc, Gtable, Qcalc and Qtable even mean in this context?

A two-level system is in a quantum state = α₁₁ + a22, which can be represented by the vector a = {a1, a2}. We are looking for the conditions under which is eigenstate of the operator c., defined below. 1) In the expression c.ỗ, c is a vector with components {Cr, Cy, Cz} ( C; are real numbers) and ỗ is a vector with components {σx, σy,σz} (σ¿ are 2 × 2 matrices). This means that the operator c. can also be represented by a 2 × 2 matrix. Write the matrix of the operator c. knowing that 0 0x = = (₁ }) = ( 5 ) . Oy σz= 0 (6-99) 1 0 1 0 2) In matrix form, the eigenvalue equation is AX AX, where A is the matrix of the operator of interest, X the column matrix representing the eigenvector and the corresponding eigenvalue. Write the eigenvalue equa- tion that needs to be verified for the quantum state to be an eigenvector of the operator c.. = 3) Note that AX = XX ⇒ (A - I)X 0, where I is the identity matrix. (AAI)X=0 is true if and only if det(A - I) = 0. Solve this equation for the operator…