Momentum-Parity, ground state quantum no even-even odd-odd

In summary, the nucleus with the ground state of ##1+## has an extra proton and neutron. All even-even nuclei must have a ground state of ##0+##. The strong force is not a nice 1/r^2 potential, while you still get some shell-like structure all different orbitals within those shells have different energies. As a result, they are always filled in pairs (spin up/down), so every pair of protons (and neutrons) will cancel each other in terms of spin contribution.
  • #1
binbagsss
1,254
11
Notation: ##J^p## - ##J## the total angular momentum, ##p## the parity = ##+## or ## -. ##

Ok so I'm given a diagram of energy levels, all have a ground state of ##0+##, except one which has a ground state of ##1+##.
I'm askeed to indentify which set of energy levels belong to which nucleus, the nuclei are all even-even except Fe : ##A=18, Z=9##,
So the answer is that ## 1+ ## ground state is only possible from Fe, and that all even-even nuclei must have a ground state of ##0+##.

Questions:
- I don't understand why ANY even-even nuclei has ##J^p=0^+##. I know that a filled shell has ##J=0 ##So this is fine for double-magic nucleus, but not for a magic, or general even-even nucleus.

- I don't understand the addition of angular momenta given by my textbook :
Because both neutrons and protons have one nucleon in the ##1d^{5/2}## level, the addition of angular momenta is: ##J=5/2+5/2=5,4,3,2,1##*

What I don't understand is which angular momenta you are supposed to be adding so here there's only one extra proton/neutron, but in general, for a odd-odd nuclei say you had 3 extra protons in a energy level, say it has ##J=J_{p} ##and 1 extra neutron in a energy level, say J##=J_{n} ##, then how should the addition work? Would you add ##J_{p}## and ##J_{n} ## in accord to * ? how would you account for the fact there are 3 extra protons but only one extra neutron?

- In terms of addition of angular momentum, how do you see it sums to ##0 ##for an even-even nuclei. I see that as magic numbers are even, there will be an even number of surplus nucleons in a energy level for non double-magic numbers. Does this somehow differ the addition of ##J## compared to odd-odd so that 0 is possible?

Thanks any help really appreciated.
 
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  • #2
The strong force is not a nice 1/r^2 potential, while you still get some shell-like structure all different orbitals within those shells have different energies. As a result, they are always filled in pairs (spin up/down), so every pair of protons (and neutrons) will cancel each other in terms of spin contribution.
At most, you have one proton and one neutron to consider for the ground-state.
 
  • #3
mfb said:
..will cancel each other in terms of spin contribution.
.

So it is assumed that ##l=0##? (The question does not say.)
 

1. What is momentum-parity in quantum mechanics?

Momentum-parity is a quantum number that describes the symmetry of a system under reflections. It determines whether a system is symmetric or anti-symmetric under spatial inversions.

2. How is ground state defined in quantum mechanics?

Ground state is the lowest energy state of a quantum system. It is the state in which the system is most stable and has the lowest possible energy.

3. What does it mean for a quantum system to be even-even?

In quantum mechanics, an even-even system is one in which both the number of protons and neutrons are even. This results in a symmetric wave function and a ground state with zero angular momentum.

4. What is the significance of odd-odd systems in quantum mechanics?

An odd-odd system is one in which both the number of protons and neutrons are odd. This results in an anti-symmetric wave function and a non-zero ground state angular momentum. These systems have unique properties and are important in many areas of physics, including nuclear and atomic physics.

5. How does the ground state of an even-even system differ from that of an odd-odd system?

The ground state of an even-even system has zero angular momentum, while the ground state of an odd-odd system has a non-zero angular momentum. This leads to different energy levels and properties for these two types of systems.

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