# The quantum number n and its restrictions

• pondzo
In summary, the process of solving the Schrodinger wave equation for the hydrogen atom introduces a constant ##\lambda## which must be an integer in order for the power series to have a finite power. This constraint arises because the wave function must be normalisable. The quantum number n arises as a combination of the angular momentum quantum number l and the excitation level k, where k is an integer determined by the number of radial nodes in the wave function.
pondzo
So in class we went through the process of solving the S.W.E for the hydrogen atom.

During the process a constant ##\lambda_n=\frac{ze^2}{4\pi\epsilon_0\hbar}(\frac{\mu}{2|E_n|})^{\frac{1}{2}}## is introduced, where mu represents the reduced mass of the electron.

Later this constant is put on the following restriction so that the power series has a finite power; ##\lambda_n=l+k+1## (where k is the index number for the power series).

It then follows that this constant must be an integer since l and k are. ##\lambda_n=n\geq l+1##.

I would like to know the physical reason why the quantum number n arises and why it is put under the constraint ##n\geq l+1##.

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Maybe I am not remembering correctly But I think it is n that is the integer, not lambda. The constant lambda is a physical quantity with physical units.

Asking "why" in physics is a troublesome thing. It depends on what kind of answer you will be happy with.

At one level, the answer is what you have already done in class. It arises because of the various boundary conditions and the need to solve the equation.

At another level it is a physical continuity and symmetry kind of thing. In essence, the wave function has to fit into a "mode" of rotational symmetry, for example. And it has to satisfy certain restrictions in the radial direction in order to be finite and correspond to a finite probability of detecting an electron. So there are only certain possibilities. So you come to discover the atomic orbitals. And those are arranged by integers.

With only a little more sophistication you come to group theory and functional representations. To motivate that, think about a system that has mirror symmetry in the plane x=0. Such a system will naturally break into functions that are even, f(x) = f(-x), and functions that are odd, g(x) = -g(-x). Depending on the details of the system, it means you can have the system in either f or g, but not a mix. Similarly with a system with spherical symmetry. There are categories of functional characteristics that are possible, and they are characterized by a set of integers.

But fundamentally, the "why" is still because it has to satisfy the equation under the boundary conditions.

pondzo said:
So in class we went through the process of solving the S.W.E for the hydrogen atom.

During the process a constant ##\lambda_n=\frac{ze^2}{4\pi\epsilon_0\hbar}(\frac{\mu}{2|E_n|})^{\frac{1}{2}}## is introduced, where mu represents the reduced mass of the electron.

Later this constant is put on the following restriction so that the power series has a finite power; ##\lambda_n=l+k+1## (where k is the index number for the power series).

It then follows that this constant must be an integer since l and k are. ##\lambda_n=n\geq l+1##.

I would like to know the physical reason why the quantum number n arises and why it is put under the constraint ##n\geq l+1##.

The reason that ##\lambda## must be equal to an integer is so that the power series terminates. If ##\lambda## is not an integer, then you have an infinite power series and a non-normalisable wave function. There might be a bit of work to show this, but that's the reason.

Not really sure what your question means. The problem of the hydrogen atom can be separated into a radial and angular part. The angular momentum quantum number is quantized because the angular wavefunction has to be single valued at the poles. k is just the excitation level of the radial wavefunction for a fixed angular momentum l. As you increase the energy (fixing l), the number of radial nodes increases, and these states are labeled by k. Since the number of nodes has to be integral, k is an integer. And n is just l+k+1. We defined n this way because it just so happens that the different states with the same 1+l+k have about the same energy, given by Rydberg's formula. But really, k is perhaps more "fundamental" than n.

## 1. What is the quantum number n and how does it relate to an atom's energy level?

The quantum number n is one of the four quantum numbers used to describe the energy state of an atom. It represents the principal energy level, or shell, of an electron. The higher the value of n, the farther the electron is from the nucleus and the higher its energy.

## 2. What are the possible values for the quantum number n?

The quantum number n can have any positive integer value starting from 1. This means that an electron can have energy levels of 1, 2, 3, and so on.

## 3. What is the significance of the restrictions on the quantum number n?

The restrictions on the quantum number n help to determine the number of electrons that can occupy a particular energy level. The maximum number of electrons in a given energy level is given by the formula 2n^2, where n is the value of the quantum number. This helps to explain why certain elements have a specific number of electrons in their outermost energy level.

## 4. How do the restrictions on the quantum number n affect the electron configuration of an atom?

The restrictions on the quantum number n play a crucial role in determining the electron configuration of an atom. The electron configuration is the arrangement of electrons in the different energy levels of an atom. The restrictions help to determine the order in which electrons fill up the energy levels, with lower energy levels being filled first before moving on to higher energy levels.

## 5. Can there be more than one electron in the same energy level with the same value of the quantum number n?

No, according to the Pauli exclusion principle, no two electrons in an atom can have the same set of four quantum numbers. Therefore, if two electrons are in the same energy level (same value of n), they must have different values for the other quantum numbers, such as the angular momentum quantum number and the magnetic quantum number.

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