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## Main Question or Discussion Point

A question came up about deducing the number of possible energy states within a certain momentum ##p## using momentum space.

To make my question easier to understand, I deliberately chose ##p## and not a particular increment ##dp## and I assume a 2 dimensional momentum space with coordinates ##x## and ##y##. The concerning particle thus only has translational kinetic energy in these 2 coordinates.

A particle within a box of volume ##V## can have the same momentum ##p## in different directions within that box. In a 2D momentum space this momentum ##p## is therefore given by a circle with radius ##p##.

From what I understand, the number of possible energy states ##N_s## in this 2D case is then deduced from the area of the circle multiplied by the number of energy states in the ##x## and ##y## coordinates:

$$N_s = \frac{L_x \cdot p_x}{h} \cdot \frac{L_y \cdot p_y}{h} \cdot \pi$$

Where ##L## is the length of the box in a certain dimension (given by subscript ##x## or ##y##).

Here's my question regarding this formula:

I can see that the formula assumes that the density of energy states is homogenous over the circular p-space because it is merely multiplying the number of energy states in the ##x## dimension by the number of energy states in the ##y## dimension. However, I don't understand why this is the case, because from what I know, the number of possible energy states in a certain direction is proportional to the length of the box in that very same direction. If a certain momentum has a combined ##x## and ##y## direction, shouldn’t the number of possible energy states within that momentum vector be dependent on the length of the box in that same direction and not by the ##x## and ##y## coordinates seperataly?

To make my question easier to understand, I deliberately chose ##p## and not a particular increment ##dp## and I assume a 2 dimensional momentum space with coordinates ##x## and ##y##. The concerning particle thus only has translational kinetic energy in these 2 coordinates.

A particle within a box of volume ##V## can have the same momentum ##p## in different directions within that box. In a 2D momentum space this momentum ##p## is therefore given by a circle with radius ##p##.

From what I understand, the number of possible energy states ##N_s## in this 2D case is then deduced from the area of the circle multiplied by the number of energy states in the ##x## and ##y## coordinates:

$$N_s = \frac{L_x \cdot p_x}{h} \cdot \frac{L_y \cdot p_y}{h} \cdot \pi$$

Where ##L## is the length of the box in a certain dimension (given by subscript ##x## or ##y##).

Here's my question regarding this formula:

I can see that the formula assumes that the density of energy states is homogenous over the circular p-space because it is merely multiplying the number of energy states in the ##x## dimension by the number of energy states in the ##y## dimension. However, I don't understand why this is the case, because from what I know, the number of possible energy states in a certain direction is proportional to the length of the box in that very same direction. If a certain momentum has a combined ##x## and ##y## direction, shouldn’t the number of possible energy states within that momentum vector be dependent on the length of the box in that same direction and not by the ##x## and ##y## coordinates seperataly?