Question

I am having issues understanding why +1, -1 are considered 2 separate dimensions of state in the context of spin. Since, the values lie on a 1 dimensional number line. For example, the number line of integers (i.e. { -1, +1 } ∈ Z) which extends from - infinity to + infinity.

It seems that the arguments being made for it to be 2 distinct dimensions say that it's because the s[om cam attain 2 distinct values that are measurably and verifiably not the same one. Although, the concept of a magnitude (to attain the other values on the number line) has not yet been introduced, in the source that I'm following anyway.

But, does this then mean that a 1D number line like the ones that extend from -infinity to + infinity or even from 0 to + infinity are not 1-dimensional at all and rather, have infinite dimensions ? And, does a mathematical dice then have 6 dimensions ?

Answer 1

In the context of spin, +1 and -1 are considered two separate dimensions of state because they refer to the spin quantum number, which corresponds to the z component of the spin. A 1D number line representing a continuous range of values is still considered one-dimensional. A mathematical dice does not have six dimensions; the six sides represent different outcomes or states.

Step-by-step explanation:

In the context of spin, +1 and -1 are considered two separate dimensions of state because they refer to the spin quantum number, which corresponds to the z component of the spin.

The z component can have two distinct values, +1/2 and -1/2, representing the two possible spin states of an electron.

While these values lie on a 1D number line, they represent distinct dimensions because they indicate different magnitudes and directions of the electron's spin.

On the other hand, a 1D number line that extends from -∞ to +∞ or from 0 to +∞ is still considered one-dimensional because it represents a continuous range of values along a single axis.

The concept of dimensions in physics is related to the number of independent variables required to fully describe a physical system

A mathematical dice, which has six sides, is not considered to have six dimensions in the same sense. It is a 1D object (line) extended into 3D space.

The six sides represent different possible outcomes or states of the dice, not dimensions.