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For all I know, an one-dimensional free particle has 1 degree of freedom and 3 degrees of freedom in the 3-D world. And in thermal physics, one-dimensional simple harmonic oscillator has 2 degrees of freedom, it is a different concept with the "degrees of freedom" in mechanics.

However, my professor who teaches quantum mechanics said that there are 2 degrees of freedom for an one-dimensional free particle and they are $x$ and $p$ (i.e. position and momentum) respectively. Then according to this, he explained that one atom in a solid has 6 (there are 3 directions i.e.$x$, $y$, $z$ and 2 degrees of freedom for each direction) degrees of freedom. And he still sticks to his own opinion.

But no book has ever said that like my teacher. I'm getting confused. So I hope someone could explain the concept of degrees of freedom to me.

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There seem to be two different notions of degrees of freedom here.

The first, and usual, is the dimension of the configuration space of the system, that is, the number of variables needed to uniquely specify its position/shape/configuration (not it's motion). This is what one calls degrees of freedom in almost every context.

The second, and non-standard, is the dimension of the phase space, which, in the absence of constraints/gauge symmetries, simply consists of all the configuration variables and their canonical momenta, i.e. its dimension is twice the degrees of freedom.

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  • $\begingroup$ So what my teacher said maybe in phase space? $\endgroup$ – Wang Yun Mar 7 '16 at 11:29
  • $\begingroup$ By the way, can I use Chinese when I ask one question? $\endgroup$ – Wang Yun Mar 7 '16 at 11:29
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    $\begingroup$ @StephenWong: Yes, I think your teacher meant dimension of the phase space or number of initial conditions to solve the equations of motion, not "degrees of freedom". And no, the language of physics.SE is exclusively English. $\endgroup$ – ACuriousMind Mar 7 '16 at 11:32

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