Degree of Freedom: Explaining n,l & m Quantum Numbers

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The discussion centers on the quantum numbers n, l, and m, which describe an electron's state in a hydrogen atom and represent its degrees of freedom in three-dimensional space. The quantum number n indicates the energy level, while l and m specify the angular momentum and its direction, respectively. Degrees of freedom refer to the number of independent parameters that define a system's state, aligning with the roles of n, l, and m. The quantum numbers arise from solving the hydrogen Schrödinger equation, providing a framework for understanding electron behavior. Overall, the relationship between these quantum numbers and degrees of freedom illustrates the complexity of electron dynamics in atomic structures.
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An electron moves around a nucleus in 3d space so it is required that there should be 3 quantum no. each specifying its degree of freedom .

the three quantum no. are n,l & m , but n,l& m specify the energy levels avalaible to electron & m specifies the direction of angular momentum vector .

So what is actually meant by degree of freedom & how do n,l & m specify it ?
 
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i'm not sure what you are asking, do you want a claasical explenation of what n,l and m mean?

n,l and m simply come from solving the hydrogen Schödinger equation. And from wikipedia

Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters. In mathematical terms, the degrees of freedom are the dimensions of a phase space.

so this definition agrees with what n,l and m does, i guess.
 
I studied my self and found the ans from hydrogen atom
 

Thread 'Two equivalent statements of time reversal symmetric Hamiltonian'

Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?
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