Bosons and Fermions
Dear all,
I appologize for the delay. Here an update of our lecture notes, which includes a short description of quantum many-body-states for non-interacting particles, which form the basis for quantum gases.
Lecture Notes (starting at page 64)
There, you can also find a question and two excercises (written in blue text). Please, consider the exercises as homework assignment, due for next Monday. I am aware, that many of you are struggeling with the their modular project. So please tell me if you have problems to find the time for doing the homework.
I appologize for the delay. Here an update of our lecture notes, which includes a short description of quantum many-body-states for non-interacting particles, which form the basis for quantum gases.
Lecture Notes (starting at page 64)
There, you can also find a question and two excercises (written in blue text). Please, consider the exercises as homework assignment, due for next Monday. I am aware, that many of you are struggeling with the their modular project. So please tell me if you have problems to find the time for doing the homework.
Good morning, professor. Excuse me, I have some doubts about the topic.
ResponderBorrarThe first one is concerning to the equation on page 67, is the main difference giben by the operators PI_sym and PI_asym is the type of micro-states that we talk about?
For the first part of the last excercise on page 68, are we going to use | j1=2, j2=3 ..> for it?
Could we have a little bit more information about the occupation number? I mean, how can we relate it with the theory of ensembles?
Thank you.
Respondiendo a la primera pregunta: Los sistemas con partículas indistinguibles, los podemos encontrar solamente en estados simetrizados (eso es el caso de bosones) o anti-simetrizados (en el caso de fermiones). Un micro-estado simetrizado (para bosones) queda igual cuando intercambiamos dos partículas arbitrarias en el sistema. Un micro-estado anti-simetrizado (para fermiones) cambia de signo cuando intercambiamos dos partículas arbitrarias en el sistema.
ResponderBorrarTodas las partículas elementales y todas las partículas compuestos por ellas pueden clasificarse como bosones o fermiones (fotones son bosones; electrones, protones, neutrones son fermiones; dos fermiones juntos forman un boson, etc.). Entonces, si tenemos un sistema compuesto por N partículas de este típo, los micro-estados deben ser los simetrizados o los anti-simetrizados, dependiendo del tipo de partícula que tenemos.
Sorry, I just realize that I should have anwered in English
Second question: Please read carefully, the statement of the problem. There, it says that each particle maybe found in only two states. So for instance j2 indicates the state of the second particle. If there are only two states, thant j2 can be either "0" or "1".
Third question: Reading your question, I guess that you understand how to find the occupation numbers for a given many-particle micro-state. You just want to know, why they are useful for calculating the partition function of such quantum systems. The excercise on page 68 is meany to show precisely that -- at least in the case of bosons, you will see that there are many configurations which are different (for distinguishable particles) which however become the same state after symmetrization. But a system of indistinguishable particles can only be found in symmetrized states, so for calculating the partition function we have to count only those. Now, you should see from the excercise that the occupation numbers are the same if the symmetrized state is the same and otherwise they are different. That means that in order to sum over all symmetrized states, we can simply sum over all different combinations of occupation numbers.
Clearly, in the lecture notes there is still a gap between the description of the symmetrized many-particle quantum states (micro-states) and the calculation of partition functions for a quantum gas. I will try to fill up this gap until Friday.