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TAMAL DEY
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Why is it necessary that the principal quantum number of a Hydrogen atom problem in Quantum mechanics must be an integer?Couldn't it be any fraction?
TAMAL DEY said:Why is it necessary that the principal quantum number of a Hydrogen atom problem in Quantum mechanics must be an integer?Couldn't it be any fraction?
Thor90 said:In quantum mechanics a quantum number is "label" to address an eigenvalue of some conserved quantity (that in QM formalism is an operator that commutes with the hamiltonian), and the principale ones, for the classic problem of an electon that orbits around a nucleous, are the eigenvalues of the Hamiltonian itself (the energy), of the angular momentum, the magnetization and the spin operator.
So it makes sanse to say that n (the principal quantum number) is equal to 1, since it says that the electon is in the first excited state, so the second smallest eigenvalue of the hamiltonian (which will have his own value in some unit of misure of the energy). On the contrary, it makes no sense at all to say that n=3/2 or some other fraction since it's simply a label from the lower state (the Hamiltonian must always be bound from below) to the infinity.
The principal quantum number, denoted by the letter n, represents the energy level or shell that an electron occupies in an atom. It determines the size and energy of an electron's orbital and helps in predicting the electron's behavior. The fact that it is an integer signifies that the energy levels in an atom are discrete and not continuous.
The principal quantum number cannot be negative because it represents the energy level of an electron in an atom, and energy levels cannot be negative. Additionally, it cannot be a fraction or decimal because it signifies the number of nodes or regions where an electron cannot exist within an orbital. These nodes can only exist at discrete values, which are the positive integers.
The principal quantum number determines the amount of energy an electron has in an atom. As the principal quantum number increases, the energy level of the electron also increases. This means that electrons in higher energy levels have more energy and are further away from the nucleus compared to electrons in lower energy levels.
The maximum value of the principal quantum number is determined by the number of electrons in an atom. It is equal to the number of energy levels or shells in an atom. For example, in an atom with 6 electrons, the maximum value of n is 6, as there are only 6 possible energy levels for these electrons to occupy.
No, the principal quantum number must be unique for each electron in an atom. This is because the principal quantum number not only represents the energy level of an electron but also its other quantum properties such as angular momentum and magnetic moment. As no two electrons in an atom can have the same set of quantum numbers, the principal quantum number must also be different for each electron.