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Ever two-particle wave function is a product of two one-particle wave functions.
Is this true?
if not, can you give me a example ?
Thank you :)
Is this true?
if not, can you give me a example ?
Thank you :)
If that were true, then any function in 2 variables can be decomposed as a product of two functions in one variable.Ever two-particle wave function is a product of two one-particle wave functions.
Is this true?
if not, can you give me a example ?
Thank you :)
It's not true any time those two particles are interacting. If your wave function is a product of two single-particle functions, then every observable quantity is a product of the probabilities for the respective particles. I.e. they're statistically uncorrelated and so, independent of each other.Ever two-particle wave function is a product of two one-particle wave functions. Is this true? if not, can you give me a example ?
Is the converse true? That is, if they're statistically uncorrelated, can they be written as a direct product?It's not true any time those two particles are interacting. If your wave function is a product of two single-particle functions, then every observable quantity is a product of the probabilities for the respective particles. I.e. they're statistically uncorrelated and so, independent of each other.
Statistical non-correlation means that P(A|B) = P(A)P(B), so yes.Is the converse true? That is, if they're statistically uncorrelated, can they be written as a direct product?
oh yes, that is almost by definition. If you have P(x,y), and P(x|y=a)=P(x,a)=f(x)g(a), then it must be true that P(x,y)=f(x)g(y), or that the state can be written:Statistical non-correlation means that P(A|B) = P(A)P(B), so yes.
I might want to add to the earlier that any set of interacting (correlated) particles may still be written as a linear expansion of different single-particle functions though. (Handwaving: Throw out the interaction terms from the Hamiltonian and solve for the single particles which can be used as a basis for the interacting Hamiltonian. E.g. a Slater determinant)