QM Definition of Center of Mass

[LarryS]

The concept of "Center of Mass" is mentioned often in QM literature. I saw one blurb stating that a diatomic molecule could be represented as a SHO vibrating about the "center of mass" between the two atoms. But is there a precise definition of center of mass in QM based solely on the state function? Thanks in advance.

 

[The_Duck]

The center of mass position is an operator just like the position of a single particle. For example if you have two particles, the center of mass is (m1r1 + m2r2)/(m1 + m2) where in QM r1 and r2 get promoted to operators.

The "blurb" you mention is referring to this situation: If you have a two-particle system you could write the Schrödinger equation in terms of the positions r1 and r2 of the two particles. But you could also write it in terms of the variables R = (r1 + r2)/2 [the center of mass] and r = (r1 - r2) [the relative displacement of the two particles]. If the potential energy V is simply a function of the distance between the two particles, then the Schrödinger equation separates into two parts. One part involves only R, and describes the motion of the center of mass; this part of the equation looks like the Schrödinger equation for a single free particle of mass (m1 + m2). The other part of the equation involves only r and describes the relative motion of the two particles; this part looks like the Schrödinger equation for a single particle with mass m1m2/(m1 + m2) in a potential V. So this is useful because two-particle systems with an inter-particle potential turn out to look just like one particle in an external potential (plus the free-particle-like propagation of the center of mass).

 

[LarryS]

The center of mass position is an operator just like the position of a single particle. For example if you have two particles, the center of mass is (m1r1 + m2r2)/(m1 + m2) where in QM r1 and r2 get promoted to operators.

The "blurb" you mention is referring to this situation: If you have a two-particle system you could write the Schrödinger equation in terms of the positions r1 and r2 of the two particles. But you could also write it in terms of the variables R = (r1 + r2)/2 [the center of mass] and r = (r1 - r2) [the relative displacement of the two particles]. If the potential energy V is simply a function of the distance between the two particles, then the Schrödinger equation separates into two parts. One part involves only R, and describes the motion of the center of mass; this part of the equation looks like the Schrödinger equation for a single free particle of mass (m1 + m2). The other part of the equation involves only r and describes the relative motion of the two particles; this part looks like the Schrödinger equation for a single particle with mass m1m2/(m1 + m2) in a potential V. So this is useful because two-particle systems with an inter-particle potential turn out to look just like one particle in an external potential (plus the free-particle-like propagation of the center of mass).

Makes sense. But shouldn't your "R" variable be defined as R = (m1r1 + m2r2)/(m1 + m2) instead of R = (r1 + r2)/2?

 

[The_Duck]

Oops, yes, it should.

 

[tom.stoer]

one can even start with the two variables r1, r2 and find a unitary transformation to r, R where P separates i.e. where the two-particle system becomes a plane wave in R with conserved total momentum P as it should.

 

[StevieTNZ]

When you speak of a macroscopic objects position, we are really stating its center of mass?

Say the dot is the center of mass location, and the arrows the rest of the object spanning from that location:
<-- . -->
Can't you have another center of mass located where the <-- -->'s are located? So two macroscopic objects are overlapping?