Quantum Two 1

2

Many Particle Systems Revisited 3

4

5

6

7

8

9

10

11

12

13

14

15

Thus, we write Thus, the quantum mechanical description in the position representation involves a single wave function but it is not simply a complex number assigned to each point in space. Indeed, it is a simultaneous function of the position coordinates of all the particles in the system. 16

Thus, we write Thus, the quantum mechanical description in the position representation involves a single wave function but it is not simply a complex number assigned to each point in space. Indeed, it is a simultaneous function of the position coordinates of all the particles in the system. 17

Thus, we write Thus, the quantum mechanical description in the position representation involves a single wave function but it is not simply a complex number assigned to each point in space. Indeed, it is a simultaneous function of the position coordinates of all the particles in the system. 18

Thus, we write Thus, the quantum mechanical description in the position representation involves a single wave function but it is not simply a complex number assigned to each point in space. Indeed, it is a simultaneous function of the position coordinates of all the particles in the system. 19

Thus, we write Thus, the quantum mechanical description in the position representation involves a single wave function but it is not simply a complex number assigned to each point in space. Indeed, it is a simultaneous function of the position coordinates of all the particles in the system. 20

21

22

23

24

Comment: There is also no particular reason to work in the many particle position representation. Indeed, in the state space associated with each particle, there are many different choices of basis sets to choose from. Thus, from the momentum eigenstates we can construct the direct product basis of the many particle momentum representation, which obey and in terms of which 25

Comment: There is also no particular reason to work in the many particle position representation. Indeed, in the state space associated with each particle, there are many different choices of basis sets to choose from. Thus, from the momentum eigenstates we can construct the direct product basis of the many particle momentum representation, which obey and in terms of which 26

Comment: There is also no particular reason to work in the many particle position representation. Indeed, in the state space associated with each particle, there are many different choices of basis sets to choose from. Thus, from the momentum eigenstates we can construct the direct product basis of the many particle momentum representation, which obey and in terms of which 27

Comment: There is also no particular reason to work in the many particle position representation. Indeed, in the state space associated with each particle, there are many different choices of basis sets to choose from. Thus, from the momentum eigenstates we can construct the direct product basis of the many particle momentum representation, which obey and in terms of which 28

Comment: There is also no particular reason to work in the many particle position representation. Indeed, in the state space associated with each particle, there are many different choices of basis sets to choose from. Thus, from the momentum eigenstates we can construct the direct product basis of the many particle momentum representation, which obey and in terms of which we may expand in terms of a momentum wave function 29

30

31

32

33

34

35

36

37

38

39

40

How about particles with spin? We just follow the rules, and see what happens. For N spin-1/2 particles moving in 3D, for example, we have individual state spaces that now take into account the spin degrees of freedom, i.e., which allows for direct product states in the combined space, obeying and in terms of which we can expand an arbitrary state. 41

How about particles with spin? We just follow the rules, and see what happens. For N spin-1/2 particles moving in 3D, for example, we have individual state spaces that now take into account the spin degrees of freedom, i.e., which allows for direct product states in the combined space, obeying and in terms of which we can expand an arbitrary state. 42

How about particles with spin? We just follow the rules, and see what happens. For N spin-1/2 particles moving in 3D, for example, we have individual state spaces that now take into account the spin degrees of freedom, i.e., which allows for direct product states in the combined space, obeying and in terms of which we can expand an arbitrary state. 43

How about particles with spin? We just follow the rules, and see what happens. For N spin-1/2 particles moving in 3D, for example, we have individual state spaces that now take into account the spin degrees of freedom, i.e., which allows for direct product states in the combined space, obeying and in terms of which we can expand an arbitrary state. 44

How about particles with spin? We just follow the rules, and see what happens. For N spin-1/2 particles moving in 3D, for example, we have individual state spaces that now take into account the spin degrees of freedom, i.e., which allows for direct product states in the combined space, obeying and in terms of which we can expand an arbitrary state. 45

How about particles with spin? We just follow the rules, and see what happens. For N spin-1/2 particles moving in 3D, for example, we have individual state spaces that now take into account the spin degrees of freedom, i.e., which allows for direct product states in the combined space, obeying and in terms of which we can expand an arbitrary state. 46

How about particles with spin? We just follow the rules, and see what happens. For N spin-1/2 particles moving in 3D, for example, we have individual state spaces that now take into account the spin degrees of freedom, i.e., which allows for direct product states in the combined space, obeying and in terms of which we can expand an arbitrary state. 47

We write: Thus, the quantum mechanical description of this system requires the specification of a wave function having components associated with the different possible spin configurations of the system: 48

We write: Thus, the quantum mechanical description of this system requires the specification of a wave function having components associated with the different possible spin configurations of the system: 49

We write: Thus, the quantum mechanical description of this system requires the specification of a wave function having components associated with the different possible spin configurations of the system: 50

We write: Thus, the quantum mechanical description of this system requires the specification of a wave function having components associated with the different possible spin configurations of the system: 51

Thus, although it may get notationally more complicated, there is no real impediment to applying the postulates of quantum mechanics to the state spaces and state vectors of systems of many particles, even those with internal (e.g., spin) degrees of freedom. In this last example, the basic rules governing the construction of the direct product space imply that all operators (spin and spatial) associated with any particle in the system automatically commute with those of any other particle Thus, computation of quantities of interest, while it may lead to complicated multi-dimensional integrals, is nonetheless straightforward to understand and interpret. In the next segment, we briefly consider the typical Hamiltonian and the subsequent evolution of many particle systems, focusing on the surprisingly useful case of a system of non-interacting particles. 52

Thus, although it may get notationally more complicated, there is no real impediment to applying the postulates of quantum mechanics to the state spaces and state vectors of systems of many particles, even those with internal (e.g., spin) degrees of freedom. In this last example, the basic rules governing the construction of the direct product space imply that all operators (spin and spatial) associated with any particle in the system automatically commute with those of any other particle. Thus, computation of quantities of interest, while it may lead to complicated multi-dimensional integrals, is nonetheless straightforward to understand and interpret. In the next segment, we briefly consider the typical Hamiltonian and the subsequent evolution of many particle systems, focusing on the surprisingly useful case of a system of non-interacting particles. 53

Thus, although it may get notationally more complicated, there is no real impediment to applying the postulates of quantum mechanics to the state spaces and state vectors of systems of many particles, even those with internal (e.g., spin) degrees of freedom. In this last example, the basic rules governing the construction of the direct product space imply that all operators (spin and spatial) associated with any particle in the system automatically commute with those of any other particle. Thus, computation of quantities of interest, while it may lead to complicated multi-dimensional integrals, is nonetheless straightforward to understand and interpret. In the next segment, we briefly consider the typical Hamiltonian and the subsequent evolution of many particle systems, focusing on the surprisingly useful case of a system of non-interacting particles. 54

Thus, although it may get notationally more complicated, there is no real impediment to applying the postulates of quantum mechanics to the state spaces and state vectors of systems of many particles, even those with internal (e.g., spin) degrees of freedom. In this last example, the basic rules governing the construction of the direct product space imply that all operators (spin and spatial) associated with any particle in the system automatically commute with those of any other particle. Thus, computation of quantities of interest, while it may lead to complicated multi-dimensional integrals, is nonetheless straightforward to understand and interpret. In the next segment, we briefly consider the typical Hamiltonian and the subsequent evolution of many particle systems, focusing on the surprisingly useful case of a system of non-interacting particles. 55

56