Quantum Two 1

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Evolution of Many Particle Systems 3

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Since direct products commute, we can combine all the spatial parts and all the spin parts to write this in the form where describes all the spatial variables, and describes all the spin variables. 10

Since direct products commute, we can combine all the spatial parts and all the spin parts to write this in the form where describes all the spatial variables, and describes all the spin variables. 11

Since direct products commute, we can combine all the spatial parts and all the spin parts to write this in the form where describes all the spatial variables, and describes all the spin variables. 12

Since direct products commute, we can combine all the spatial parts and all the spin parts to write this in the form where describes all the spatial variables, and describes all the spin variables. 13

Since direct products commute, we can combine all the spatial parts and all the spin parts to write this in the form where describes all the spatial variables, and describes all the spin variables. 14

Thus, we can solve the energy eigenvalue problem in, spanned by the direct product position states, without worrying about spin at all. After we are done, we can form direct products of the complete set of spatial energy eigenstates, with a complete set of spin states, for, thus obtaining a full set of solutions to the energy eigenvalue problem spanning the total direct product space. Implementing this idea, we project the eigenvalue equation onto the many particle position representation, obtaining a partial differential equation for the many-particle energy eigenfunctions 15

Thus, we can solve the energy eigenvalue problem in, spanned by the direct product position states, without worrying about spin at all. After we are done, we can form direct products of the complete set of spatial energy eigenstates, with a complete set of spin states, for, thus obtaining a full set of solutions to the energy eigenvalue problem spanning the total direct product space. Implementing this idea, we project the eigenvalue equation onto the many particle position representation, obtaining a partial differential equation for the many-particle energy eigenfunctions 16

Thus, we can solve the energy eigenvalue problem in, spanned by the direct product position states, without worrying about spin at all. After we are done, we can form direct products of the complete set of spatial energy eigenstates, with a complete set of spin states, for, thus obtaining a full set of solutions to the energy eigenvalue problem spanning the total direct product space. Implementing this idea, we project the eigenvalue equation onto the many particle position representation, obtaining a partial differential equation for the many-particle energy eigenfunctions 17

Thus, we can solve the energy eigenvalue problem in, spanned by the direct product position states, without worrying about spin at all. After we are done, we can form direct products of the complete set of spatial energy eigenstates, with a complete set of spin states, for, thus obtaining a full set of solutions to the energy eigenvalue problem spanning the total direct product space. Implementing this idea, we project the eigenvalue equation onto the many particle position representation, obtaining a partial differential equation for the many-particle energy eigenfunctions 18

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Indeed, using single particle potentials of this type it is often possible, to accurately treat using perturbation methods, more complicated two-body interactions, like the Coulomb interaction described above. Imagine for example, that all but one of the particles in the system were fixed in some well-defined set of single particle spatial states. This remaining particle could then be treated as moving in the potential field generated by all the others. This can be iteratively repeated for each particle. In this way, suitable potentials can thus be generated for each particle which, in some average sense, self-consistently take into account the states that the remaining particles actually end up in. This is precisely the goal of so-called self-consistent theories, such as the Central Potential Approximation, the Hartree-Fock approximation, and the Thomas- Fermi model. 27

Indeed, using single particle potentials of this type it is often possible, to accurately treat using perturbation methods, more complicated two-body interactions, like the Coulomb interaction described above. Imagine for example, that all but one of the particles in the system were fixed in some well-defined set of single particle spatial states. This remaining particle could then be treated as moving in the potential field generated by all the others. This can be iteratively repeated for each particle. In this way, suitable potentials can thus be generated for each particle which, in some average sense, self-consistently take into account the states that the remaining particles actually end up in. This is precisely the goal of so-called self-consistent theories, such as the Central Potential Approximation, the Hartree-Fock approximation, and the Thomas- Fermi model. 28

Indeed, using single particle potentials of this type it is often possible, to accurately treat using perturbation methods, more complicated two-body interactions, like the Coulomb interaction described above. Imagine for example, that all but one of the particles in the system were fixed in some well-defined set of single particle spatial states. This remaining particle could then be treated as moving in the potential field generated by all the others. This can be iteratively repeated for each particle. In this way, suitable potentials can thus be generated for each particle which, in some average sense, self-consistently take into account the states that the remaining particles actually end up in. This is precisely the goal of so-called self-consistent theories, such as the Central Potential Approximation, the Hartree-Fock approximation, and the Thomas- Fermi model. 29

Indeed, using single particle potentials of this type it is often possible, to accurately treat using perturbation methods, more complicated two-body interactions, like the Coulomb interaction described above. Imagine for example, that all but one of the particles in the system were fixed in some well-defined set of single particle spatial states. This remaining particle could then be treated as moving in the potential field generated by all the others. This can be iteratively repeated for each particle. In this way, suitable potentials can thus be generated for each particle which, in some average sense, self-consistently take into account the states that the remaining particles actually end up in. This is precisely the goal of so-called self-consistent theories, such as the Central Potential Approximation, the Hartree-Fock approximation, and the Thomas- Fermi model. 30

Indeed, using single particle potentials of this type it is often possible, to accurately treat using perturbation methods, more complicated two-body interactions, like the Coulomb interaction described above. Imagine for example, that all but one of the particles in the system were fixed in some well-defined set of single particle spatial states. This remaining particle could then be treated as moving in the potential field generated by all the others. This can be iteratively repeated for each particle. In this way, suitable potentials can thus be generated for each particle which, in some average sense, self-consistently take into account the states that the remaining particles actually end up in. This is precisely the goal of so-called self-consistent theories, such as the Central Potential Approximation, the Hartree-Fock approximation, and the Thomas- Fermi model. 31

Once a set of single particle potentials of this type has been found, the actual Hamiltonian can then be rewritten (exactly) in the form where it is to be hoped, now represents a small correction to the total final energy. 32

Once a set of single particle potentials of this type has been found, the actual Hamiltonian can then be rewritten (exactly) in the form where it is to be hoped, now represents a small correction to the total final energy. 33

Once a set of single particle potentials of this type has been found, the actual Hamiltonian can then be rewritten (exactly) in the form where it is to be hoped, now represents a small correction to the total final energy. 34

Once a set of single particle potentials of this type has been found, the actual Hamiltonian can then be rewritten (exactly) in the form where it is to be hoped, now represents a small correction to the total final energy. 35

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To show this, we just compute where the total energy eigenvalue is just the sum of the non-interacting single particle energies (as it is classically). 45

To show this, we just compute where the total energy eigenvalue is just the sum of the non-interacting single particle energies (as it is classically). 46

To show this, we just compute where the total energy eigenvalue is just the sum of the non-interacting single particle energies (as it is classically). 47

To show this, we just compute where the total energy eigenvalue is just the sum of the non-interacting single particle energies (as it is classically). 48

To show this, we just compute where the total energy eigenvalue is just the sum of the non-interacting single particle energies (as it is classically). 49

To show this, we just compute where the total energy eigenvalue is just the sum of the non-interacting single particle energies (as it is classically). 50

To show this, we just compute where the total energy eigenvalue is just the sum of the non-interacting single particle energies (as it is classically). 51

To show this, we just compute where the total energy eigenvalue is just the sum of the non-interacting single particle energies (as it is classically). 52

The corresponding wave function associated with such a state is then a product of the associated single particle eigenfunctions of the operators, the same result that one would find by using the process of separation of variables. For electronic systems, the (anti-symmetrized) direct product state that is closest to the actual ground state (which because of interactions is an entangled superposition of such states) is called the Hartree, or Hartree-Fock approximation. The difference between the energy of the Hartree-Fock solution and the true ground state energy is referred to as the correlation energy, and in many interacting systems it provides only a small correction to the total energy. 53

The corresponding wave function associated with such a state is then a product of the associated single particle eigenfunctions of the operators, the same result that one would find by using the process of separation of variables. For electronic systems, the (anti-symmetrized) direct product state that is closest to the actual ground state (which because of interactions is an entangled superposition of such states) is called the Hartree, or Hartree-Fock approximation. The difference between the energy of the Hartree-Fock solution and the true ground state energy is referred to as the correlation energy, and in many interacting systems it provides only a small correction to the total energy. 54

The corresponding wave function associated with such a state is then a product of the associated single particle eigenfunctions of the operators, the same result that one would find by using the process of separation of variables. For electronic systems, the (anti-symmetrized) direct product state that is closest to the actual ground state (which because of interactions is an entangled superposition of such states) is called the Hartree, or Hartree-Fock approximation. The difference between the energy of the Hartree-Fock solution and the true ground state energy is referred to as the correlation energy, and in many interacting systems it provides only a small correction to the total energy. 55

Thus, as we have seen 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 our discussion of many-particle systems, we have ignored the fact that in many systems of interest the particles are elements of classes of identical or indistinguishable particles. This empirical fact puts additional constraints on the symmetry properties of the state vectors of many particle systems. We will explore the consequences of these constraints at a later point in the semester. In the next segment we begin a discussion of approximation methods for solving eigenvalue problems, of the sort we have alluded to above. 56

Thus, as we have seen 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 our discussion of many-particle systems, we have ignored the fact that in many systems of interest the particles are elements of classes of identical or indistinguishable particles. This empirical fact puts additional constraints on the symmetry properties of the state vectors of many particle systems. We will explore the consequences of these constraints at a later point in the semester. In the next segment we begin a discussion of approximation methods for solving eigenvalue problems, of the sort we have alluded to above. 57

Thus, as we have seen 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 our discussion of many-particle systems, we have ignored the fact that in many systems of interest the particles are elements of classes of identical or indistinguishable particles. This empirical fact puts additional constraints on the symmetry properties of the state vectors of many particle systems. We will explore the consequences of these constraints at a later point in the semester. In the next segment we begin a discussion of approximation methods for solving eigenvalue problems, of the sort we have alluded to above. 58

Thus, as we have seen 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 our discussion of many-particle systems, we have ignored the fact that in many systems of interest the particles are elements of classes of identical or indistinguishable particles. This empirical fact puts additional constraints on the symmetry properties of the state vectors of many particle systems. We will explore the consequences of these constraints at a later point in the semester. In the next segment we begin a discussion of approximation methods for solving eigenvalue problems, of the sort we have alluded to above. 59

Thus, as we have seen 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 our discussion of many-particle systems, we have ignored the fact that in many systems of interest the particles are elements of classes of identical or indistinguishable particles. This empirical fact puts additional constraints on the symmetry properties of the state vectors of many particle systems. We will explore the consequences of these constraints at a later point in the semester. In the next segment we begin a discussion of approximation methods for solving eigenvalue problems, of the sort we have already alluded to above. 60

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