代做Phys 115B, Winter 2024 Review 2024代做留学生SQL语言

日期：2024-03-17 04:42

Phys 115B, Winter 2024 Review

3/13/2024

• Hydrogen atom

– Be familiar with the wavefunctions of the hydrogen atom and if given a wave-function like ψn,l,m, be able to write down the functional form. (using the book).

– What do the quantum numbers n, l, and m represent? Know the expectation values of H, L2 , and Lz for states of definite n, l, and m.

– What are the raising and lowering operators L± and what do they do to states with known angular momentum?

– What is the key relationship between the components of L? Know the commu-tators, and what it means that the components of L do not commute with each other.

– Remember the orthonormal properties of the spherical harmonics

• Spin

– What are allowed values of s, the spin of a particle? Name a spin-1/2 particle, or a spin-1 particle? Spin-0?

– For spin-1/2 particles, how do measurements of various spin components change expectations for the other components. Can you write eigenstates of Sx in terms of the eigenstates of Sy or Sz and vice versa?

– Be able to calculate or write down reasonably quickly the expectation values of S2 and Si .

• Matrix Representations

– Understand how to derive a matrix representation, as on the midterm. How should you define an appropriate basis?

– What is the matrix representation of an operator?

– What is the matrix representation of the spin-1/2 system? Be familiar with spinors and the Pauli spin matrices.

• Addition of angular momenta

– Know the possible values of the total angular momentum of a system when you add two angular momentum (fully aligned, fully anti-aligned, and then everything in between in integer steps)

– Know the difference between the coupled and uncoupled representations. It is common to use the coupled representation when you see dot products of two angular momenta. Why?

– Be familiar with the Clebsch-Gordan table and how to use it. If a problem involves multiple particles with different angular momenta (spin or oribtal) or mentions multiple spins, odds are it is an addition of angular momentum prob-lem and you will need to get out the CG table.

• Entanglement

– What is the definition of an entangled state?

– Is there an easy way to identify them? Think of the homework problem, as you won’t be asked to do anything more complicated than that.

• Identical Particles

– What are bosons and fermions? How does that relate to the spin of a particle?

– Know how bosons, fermions, and identical particles can arrange themselves into possible states

– Be able to calculate the degeneracy of a system of bosons or fermions, as we did on the homework so many times.

– Be familiar with the concept of exchange forces. When does the exchange force become small or negligible and what is the significance of that?

• Solids

– Be familiar with the models we derived in class - the free electron gas model and the Dirac comb.

– Why did we go into momentum space in deriving the free electron gas model? What is the area or volume of a box in momentum space for the free electron gas model?

– What leads to degeneracy pressure?

• Atoms

– In an exam, we’ll always mostly ignore any cross terms between electrons in an atom, so that all wavefunctions are Hydrogen-like. You should know how to adapt the hydrogen wavefunctions to a hydrogen-like wavefunction for an atom with Z ≠ 1.

– Be familiar with how electrons distribute themselves into wavefunctions, and how that leads qualitatively to the periodic table.

• Symmetries

– Understand how some transformations can be generated by observables - we explored spatial translations/momentum, rotations/angular momentum, and time/Hamiltonian.

– Why does a symmetry under some transformation imply a conservation law? What is a symmetry under a transformation?

– How do operators transform. under translations or rotations?

– Be familiar with how we used Taylor expansions to describe how momentum generated spatial translations (or L generated rotations)

• Selection Rules

– Know how to derive simple selection rules, e.g. Chapter 6.4.3

– Be familiar with the last homework problem - if a transition is mediated by an operator (like x or a dipole moment), what does that mean for allowed transitions?

– How do selection rules relate to matrix representations?