# (i,j) notation of orders of wave vector

• PBJ
In summary: So, in summary, the simplified formula shows that the peak wavelengths for the two electrons occur at energy states [10], [11], and [20] in the potential well. The i,j notation corresponds to the wavevectors of the particles and the notation goes up as 10, 11, and 20 because it represents the different energy states of the two electrons. Overall, this formula helps determine the peak amplitude of the wavefunction for the two electrons in the potential well.
PBJ
I am looking at a formula, simplified here, which says that peaks occur at wavelengths: $$\lambda_{ij}=...(i^{2}+j^{2})...$$ "where i, j correspond to the orders of the 2D wave vector"

The peak wavelengths are at $$\lambda_{10}$$,$$\lambda_{11}$$,$$\lambda_{20}$$

(i.e. where $$(i^{2}+j^{2})$$ terms are

$$1^{2}+0^{2}=1$$
$$1^{2}+1^{2}=2$$
$$2^{2}+0^{2}=4$$

I don't understand what is meant by "i,j correspond to the orders of the 2D wave vector" and why the notation goes up as 10,11,20...

I expect there is a simple explanation if someone would be kind enough to help me understand this.

In this problem it appears you are looking at two electrons (or any particles) in a potential well. Orders of the 2D wavevector, I'm assuming, are the energy states. I'm going to go out on a limb and say that you are looking for the peak amplitude of the wavefunction. The i,j notation, while somewhat confusing as this notation is generally left for complex numbers, are the wavevectors of the particles oscillating in the potential well.

Therefore the notation on the lambda symbol are the energy states of the two electrons, [10] is one electron in the first excited state and the second electron in the ground state, [11] first and second electrons in the first excited state and [20] first electron in the second excited state, second electron in the ground state.

## 1. What is the (i,j) notation of orders of wave vector?

The (i,j) notation of orders of wave vector is a way of representing the different components of a wave vector in two-dimensional space. The letters "i" and "j" represent the x and y components, respectively, of the wave vector.

## 2. How is the (i,j) notation used in wave vector calculations?

In wave vector calculations, the (i,j) notation is used to represent the magnitude and direction of a wave in two-dimensional space. The values of i and j are multiplied by the unit vectors in the x and y directions, respectively, to determine the overall magnitude and direction of the wave.

## 3. Can the (i,j) notation be used for three-dimensional waves?

No, the (i,j) notation is only used for two-dimensional waves. In three-dimensional space, the (i,j,k) notation is used, with "k" representing the z component of the wave vector.

## 4. How is the (i,j) notation related to the concept of phase velocity?

The (i,j) notation is directly related to the concept of phase velocity, as it represents the direction and magnitude of the wave's phase velocity in two-dimensional space. The phase velocity is the rate at which the phase of the wave propagates in a given direction.

## 5. How does the (i,j) notation differ from the (x,y) notation?

The (i,j) notation differs from the (x,y) notation in that it is specifically used for representing wave vectors, while the (x,y) notation is used for general coordinates in two-dimensional space. The (i,j) notation is also commonly used in physics and engineering, while the (x,y) notation is more commonly used in mathematics and geometry.

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