5.1: Fundamental Concepts of Multi-Particle Systems
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We have already seen that the instantaneous state of a system consisting of a single non-relativistic particle, whose position coordinate is x, is fully specified by a complex wavefunction ψ(x,t). This wavefunction is interpreted as follows. The probability of finding the particle between x and x+dx at time t is given by |ψ(x,t)|2dx. This interpretation only makes sense if the wavefunction is normalized such that ∫∞−∞|ψ(x,t)|2dx=1
Consider a system containing N non-relativistic particles, labeled i=1,N, moving in one dimension. Let xi and mi be the position coordinate and mass, respectively, of the ith particle. By analogy with the single-particle case, the instantaneous state of a multi-particle system is specified by a complex wavefunction ψ(x1,x2,…,xN,t). The probability of finding the first particle between x1 and x1+dx1, the second particle between x2 and x2+dx2, et cetera, at time t is given by |ψ(x1,x2,…,xN,t)|2dx1dx2…dxN. It follows that the wavefunction must satisfy the normalization condition ∫|ψ(x1,x2,…,xN,t)|2dx1dx2…dxN=1
In a single-particle system, position is represented by the algebraic operator x, whereas momentum is represented by the differential operator −iℏ∂/∂x. (See Section [s4.6].) By analogy, in a multi-particle system, the position of the ith particle is represented by the algebraic operator xi, whereas the corresponding momentum is represented by the differential operator pi=−iℏ∂∂xi.
Because the xi are independent variables (i.e., ∂xi/∂xj=δij), we conclude that the various position and momentum operators satisfy the following commutation relations: [xi,xj]=0,[pi,pj]=0,[xi,pj]=iℏδij.
Finally, if H(x1,x2,…,xN,t) is the Hamiltonian of the system then the multi-particle wavefunction ψ(x1,x2,…,xN,t) satisfies the usual time-dependent Schrödinger equation [see Equation ([etimed])] iℏ∂ψ∂t=Hψ.
Contributors and Attributions
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)