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Lecture 12 - Quantum Mechanical Considerations System of...

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Quantum Mechanical Considerations System of many non-interacting identical particles: Single particle in a potential field ( ) V r Schrödinger equation 2 2 ( ) ( ) ( ) 2 V r r E r m Two identical particles 2 2 2 2 1 1 2 2 1 1 1 2 2 2 2 2 1 ( ) ( , ) ( ) ( , ) ( , ) 2 2 ( ) 2 N N j j j V r r r V r r r E r r m m H V r m To compute the N-particle energy eigenstates, it is only necessary to solve the single particle Schrödinger equation single-particle eigen state # 2 2 ( ) ( ) ( ) 2 n n n V r u r E u r m Complete set of eigenfunctions for the N particle problem can be taken by arbitrary products of single- particle states Example 1) 1 2 3 1 1 2 2 3 3 ( , , ) ( ) ( ) ( ) eigenstate 1 1st particle r r r u r u r u r   Eigenvalue 1 2 3 E E E E 1 2 3 1 1 2 3 1 2 2 3 1 2 3 3 1 2 3 ( , , ) N H r r r E u u u u E u u u u E u Eu u u 2) 1 2 3 1 1 1 2 4 3 1 4 ( , , ) ( ) ( ) ( ) 2 r r r u r u r u r E E E In these examples, permutation of particles among the single particle states gives the same energy eigenvalue. In example (1) N! permutations.

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Symmetry Properties Quantum Field Theory: Spin integer particles (Bosons) --- symmetric for 2 particle interchange Spin half-integer particles (Fermions) --- antisymmetric Bosons Symmetry 1 2 3 2 1 3 1 3 2 ( , , ) ( , , ) ( , , ) etc. r r r r r r r r r Symmetrize Permutations 1 2 3 1 1 4 2
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Lecture 12 - Quantum Mechanical Considerations System of...

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