Why particle in a box is considered as 1-dimensional ?

[dying_star]

I'm new to the field of quantum mechanics and I have got this basic doubt.
The position of the particle in a box is defined by the sine curve. Any position on a sine curve requires 2 coordinates to get the location of the particle having the highest probability at any point of time.
But why is it considered as single dimension ?
(The particle is free to move in the left or right direction, but as we define it by a sine curve it can also move up and down.)

 

[Borek]

No, it doesn't move up and down. It moves only left and right, and solution doesn't tell you what is particle's position in terms of (x,y), but what is probability of finding it along x axis.

 

[dying_star]

Do you mean that even if the particle moves in space inside the box, we are interested to know the position with respect top X -axis.

 

[Borek]

It is like a bead on a string - it can move only left and right. Thus only x axis.

 

[AJ Bentley]

The vertical axis of the graph is the probability amplitude, not the spatial direction.
It's a graph, not a picture of the box.

It's the same as if you made a graph of the speed of a car starting down a road and stopping. The graph goes up and down - but the car doesn't. The road is one-dimensional.

 

[zzzoak]

If it is used 1-dimentional hamiltonian then the solution depends on 1 coordinate x.
If 3-dimentional then f(x,y,z) is obtained.
It depends on the potential energy if the solution is the product of separate functions
f(x,y,z)=f1(x)f2(y)f3(z).
Sometimes the form of separate functions is the same and
f(x,y,z)=g(x)g(y)g(z).
So if g(x) is known the 3-dimensional function is known too.
It depends on the definite problems under consideration.