okay this isnt exactly me asking how to solve a question but just to verify the theory in my head. i need to calculate the 4 lowest energy levels for a 3d box (that has sides (2a,a,a)) and show all the quantum numbers (i imagine it means the x,y,z components of n, l, m_{l}) and give the degeneracy values. also note the particles are spin-less. this i can do but what i dont know is, considering there is no external magnetic field does the values of m_{l} degenerate into a single energy value? Basically do i count m_{l} as a quantum number that affects degeneracy? cause so far i have counted the variations of the n components and l components. i.e if i have n(1,1,2) then the l values are(0,0,0 or 1) which results in 2 possibilities so degeneracy = 2. so do i count the m_{l}? also the next question asks about if 3 identical non-interacting spin-less particles are confined in this box find the lowest total energy for these particles. how would this work? i thought pauli exclusion principle prevented particles with identical quantum numbers being in the same state? and if all the qm numbers are the same it would have to be in the same state right? thanks for any help
You are talking about a 3D rectangular box, so why you are using the quantum number of a Hydrogen atom? Quantum number is just a representing number of the system state and they are associated with different wavefunction for different system. Ground state means the 'allowed' state with the minimum energy. Here, the 'allowed state' means state satisfying the Pauli exclusion principle. Also, please do not mix the concept of quantum number and the energy.