1. Quantum One

3. Postulate III

4. The Measurement Process
Postulate III
The Measurement Process

5. In the last lecture, we investigated the conditions under which two or more observables can have none, some, or a complete set of common eigenstates.
We saw that to have any common eigenstates at all, it is necessary that the commutator of the two observables have eigenvectors with eigenvalue zero.
For two operators that commute, this is automatically satisfied, and in that case we can always construct an ONB of simultaneous eigenstates common to them, or to any set of mutually commuting observables.
When we have enough commuting observables, so that the resulting basis vectors are uniquely labeled, we have found a complete set of commuting observables.
In this lecture, we begin discussing the 3rd postulate, which describes what happens when an arbitrary observable 𝐴 is measured on the system when it is in an arbitrary state |𝜓〉 at the time of measurement.

6. We first note first that since 𝐴 is an observable it has, by definition, a set of eigenvalues {𝑎} and an ONB of eigenvectors
which satisfy orthonormality and completeness relations
in terms of which we can expand the state on which a measurement will be made

7. We first note first that since 𝐴 is an observable it has, by definition, a set of eigenvalues {𝑎} and an ONB of eigenvectors
which satisfy orthonormality and completeness relations
in terms of which we can expand the state on which a measurement will be made

8. We first note first that since 𝐴 is an observable it has, by definition, a set of eigenvalues {𝑎} and an ONB of eigenvectors
which satisfy orthonormality and completeness relations
in terms of which we can expand the state on which a measurement will be made

9. We first note first that since 𝐴 is an observable it has, by definition, a set of eigenvalues {𝑎} and an ONB of eigenvectors
which satisfy orthonormality and completeness relations
in terms of which we can expand the state on which a measurement will be made

10. We first note first that since 𝐴 is an observable it has, by definition, a set of eigenvalues {𝑎} and an ONB of eigenvectors
which satisfy orthonormality and completeness relations
in terms of which we can expand the state on which a measurement will be made

11. We first note first that since 𝐴 is an observable it has, by definition, a set of eigenvalues {𝑎} and an ONB of eigenvectors
which satisfy orthonormality and completeness relations
in terms of which we can expand the state on which a measurement will be made

12. where the expansion coefficients are just given by the inner products
Now, the completeness relation can be written in the form
where, for fixed 𝑎,
is the sum of the orthogonal projectors onto the basis vectors that span the eigenspace 𝑆 𝑎 of 𝐴 with eigenvalue 𝑎. It is thus the total projector on to that subspace.

13. where the expansion coefficients are just given by the inner products
Now, the completeness relation can be written in the form
where, for fixed 𝑎,
is the sum of the orthogonal projectors onto the basis vectors that span the eigenspace 𝑆 𝑎 of 𝐴 with eigenvalue 𝑎. It is thus the total projector on to that subspace.

14. where the expansion coefficients are just given by the inner products
Now, the completeness relation can be written in the form
where, for fixed 𝑎,
is the sum of the orthogonal projectors onto the basis vectors that span the eigenspace 𝑆 𝑎 of 𝐴 with eigenvalue 𝑎. It is thus the total projector on to that subspace.

15. where the expansion coefficients are just given by the inner products
Now, the completeness relation can be written in the form
where, for fixed 𝑎,
is the sum of the orthogonal projectors onto the basis vectors that span the eigenspace 𝑆 𝑎 of 𝐴 with eigenvalue 𝑎. It is thus the total projector on to that subspace.

16. where the expansion coefficients are just given by the inner products
Now, the completeness relation can be written in the form
where, for fixed 𝑎,
is the sum of the orthogonal projectors onto the basis vectors that span the eigenspace 𝑆 𝑎 of 𝐴 with eigenvalue 𝑎. It is thus the total projector on to that subspace.

17. We can therefore write
where
is just the part of the state |𝜓〉 lying in the eigenspace 𝑆 𝑎 .
With these definitions, then we state the first part of the third postulate as it applies to an observable with a discrete spectrum, as follows:

18. We can therefore write
where
is just the part of the state |𝜓〉 lying in the eigenspace 𝑆 𝑎 .
With these definitions, then we state the first part of the third postulate as it applies to an observable with a discrete spectrum, as follows:

19. We can therefore write
where
is just the part of the state |𝜓〉 lying in the eigenspace 𝑆 𝑎 .
With these definitions, then we state the first part of the third postulate as it applies to an observable with a discrete spectrum, as follows:

20. Postulate III (a) - (Values obtained during measurement) The only value which can be obtained as a result of measuring an observable 𝐴 of a quantum mechanical system at a time when the system is in the normalized state |𝜓〉, is one of the eigenvalues in the spectrum of 𝐴.
Exactly which eigenvalue will be obtained cannot generally be predicted.
It is possible, however, to predict the probability for obtaining each eigenvalue of 𝐴.
The probability 𝑃(𝑎) that a measurement will yield the discrete eigenvalue 𝑎 of 𝐴 is given by the expression
where is the projector onto the eigenspace of 𝐴 associated with that eigenvalue.

21. Postulate III (a) - (Values obtained during measurement) The only value which can be obtained as a result of measuring an observable 𝐴 of a quantum mechanical system at a time when the system is in the normalized state |𝜓〉, is one of the eigenvalues in the spectrum of 𝐴.
Exactly which eigenvalue will be obtained cannot generally be predicted.
It is possible, however, to predict the probability for obtaining each eigenvalue of 𝐴.
The probability 𝑃(𝑎) that a measurement will yield the discrete eigenvalue 𝑎 of 𝐴 is given by the expression
where is the projector onto the eigenspace of 𝐴 associated with that eigenvalue.

22. Postulate III (a) - (Values obtained during measurement) The only value which can be obtained as a result of measuring an observable 𝐴 of a quantum mechanical system at a time when the system is in the normalized state |𝜓〉, is one of the eigenvalues in the spectrum of 𝐴.
Exactly which eigenvalue will be obtained cannot generally be predicted.
It is possible, however, to predict the probability for obtaining each eigenvalue of 𝐴.
The probability 𝑃(𝑎) that a measurement will yield the discrete eigenvalue 𝑎 of 𝐴 is given by the expression
where is the projector onto the eigenspace of 𝐴 associated with that eigenvalue.

23. Postulate III (a) - (Values obtained during measurement) The only value which can be obtained as a result of measuring an observable 𝐴 of a quantum mechanical system at a time when the system is in the normalized state |𝜓〉, is one of the eigenvalues in the spectrum of 𝐴.
Exactly which eigenvalue will be obtained cannot generally be predicted.
It is possible, however, to predict the probability for obtaining each eigenvalue of 𝐴.
The probability 𝑃(𝑎) that a measurement will yield the discrete eigenvalue 𝑎 of 𝐴 is the expectation value taken with respect to the state |𝜓〉
where is the projector onto the eigenspace of 𝐴 associated with that eigenvalue.

24. Postulate III (a) - (Values obtained during measurement) The only value which can be obtained as a result of measuring an observable 𝐴 of a quantum mechanical system at a time when the system is in the normalized state |𝜓〉, is one of the eigenvalues in the spectrum of 𝐴.
Exactly which eigenvalue will be obtained cannot generally be predicted.
It is possible, however, to predict the probability for obtaining each eigenvalue of 𝐴.
The probability 𝑃(𝑎) that a measurement will yield the discrete eigenvalue 𝑎 of 𝐴 is the expectation value taken with respect to the state |𝜓〉
of the projector onto the eigenspace of 𝐴 associated with that eigenvalue.

25. A geometric interpretation arises from the observation that
is obviously Hermitian, and that it is a projector, so
Thus, we can write
which implies that the probability to obtain the eigenvalue 𝑎, is just the squared length
of the part of |𝜓〉 that lies inside the eigenspace 𝑆 𝑎 .

26. A geometric interpretation arises from the observation that
is obviously Hermitian, and that it is a projector, so
Thus, we can write
which implies that the probability to obtain the eigenvalue 𝑎, is just the squared length
of the part of |𝜓〉 that lies inside the eigenspace 𝑆 𝑎 .

27. A geometric interpretation arises from the observation that
is obviously Hermitian, and that it is a projector, so
Thus, we can write
which implies that the probability to obtain the eigenvalue 𝑎, is just the squared length
of the part of |𝜓〉 that lies inside the eigenspace 𝑆 𝑎 .

28. A geometric interpretation arises from the observation that
is obviously Hermitian, and that it is a projector, so
Thus, we can write
which implies that the probability to obtain the eigenvalue 𝑎, is just the squared length
of the part of |𝜓〉 that lies inside the eigenspace 𝑆 𝑎 .

29. A geometric interpretation arises from the observation that
is obviously Hermitian, and that it is a projector, so
Thus, we can write
which implies that the probability to obtain the eigenvalue 𝑎, is just the squared length
of the part of |𝜓〉 that lies inside the eigenspace 𝑆 𝑎 .

30. A geometric interpretation arises from the observation that
is obviously Hermitian, and that it is a projector, so
Thus, we can write
which implies that the probability to obtain the eigenvalue 𝑎, is just the squared length
of the part of |𝜓〉 that lies inside the eigenspace 𝑆 𝑎 .

31. A geometric interpretation arises from the observation that
is obviously Hermitian, and that it is a projector, so
Thus, we can write
which implies that the probability to obtain the eigenvalue 𝑎, is just the squared length
of the part of |𝜓〉 that lies inside the eigenspace 𝑆 𝑎 .

32. A geometric interpretation arises from the observation that
is obviously Hermitian, and that it is a projector, so
Thus, we can write
which implies that the probability to obtain the eigenvalue 𝑎, is just the squared length
of the part of |𝜓〉 that lies inside the eigenspace 𝑆 𝑎 .

33. Note that, in this discrete case, if the eigenvalue 𝑎 is nondegenerate, so that there is only one linearly independent basis vector |𝑎〉, then the projector
onto the subspace 𝑆 𝑎 is just the projector onto this one state.
In this limit, 𝑃(𝑎) reduces to the one we obtained within Schrödinger's postulates, namely,
where is the associated expansion coefficient for the state |𝜓〉 in the basis of eigenstates of A.
Thus, the probability reduces to the squared magnitude of the associated amplitude, exactly as we asserted in Schrödinger's mechanics.

34. Note that, in this discrete case, if the eigenvalue 𝑎 is nondegenerate, so that there is only one linearly independent basis vector |𝑎〉, then the projector
onto the subspace 𝑆 𝑎 is just the projector onto this one state.
In this limit, 𝑃(𝑎) reduces to the one we obtained within Schrödinger's postulates, namely,
where is the associated expansion coefficient for the state |𝜓〉 in the basis of eigenstates of A.
Thus, the probability reduces to the squared magnitude of the associated amplitude, exactly as we asserted in Schrödinger's mechanics.

35. Note that, in this discrete case, if the eigenvalue 𝑎 is nondegenerate, so that there is only one linearly independent basis vector |𝑎〉, then the projector
onto the subspace 𝑆 𝑎 is just the projector onto this one state.
In this limit, 𝑃(𝑎) reduces to
where is the associated expansion coefficient for the state |𝜓〉 in the basis of eigenstates of 𝐴.
Thus, the probability reduces to the squared magnitude of the associated amplitude, exactly as we asserted in Schrödinger's mechanics.

36. Note that, in this discrete case, if the eigenvalue 𝑎 is nondegenerate, so that there is only one linearly independent basis vector |𝑎〉, then the projector
onto the subspace 𝑆 𝑎 is just the projector onto this one state.
In this limit, 𝑃(𝑎) reduces to
where is the associated expansion coefficient for the state |𝜓〉 in the basis of eigenstates of 𝐴.
Thus, the probability reduces to the squared magnitude of the associated amplitude, exactly as we asserted in Schrödinger's mechanics.

37. Note that, in this discrete case, if the eigenvalue 𝑎 is nondegenerate, so that there is only one linearly independent basis vector |𝑎〉, then the projector
onto the subspace 𝑆 𝑎 is just the projector onto this one state.
In this limit, 𝑃(𝑎) reduces
where is the associated expansion coefficient for the state |𝜓〉 in the basis of eigenstates of 𝐴.
Thus, the probability reduces to the squared magnitude of the associated amplitude, exactly as we asserted in Schrödinger's mechanics.

38. Note that, in this discrete case, if the eigenvalue 𝑎 is nondegenerate, so that there is only one linearly independent basis vector |𝑎〉, then the projector
onto the subspace 𝑆 𝑎 is just the projector onto this one state.
In this limit, 𝑃(𝑎) reduces to
where is the associated expansion coefficient for the state |𝜓〉 in the basis of eigenstates of 𝐴.
Thus, the probability reduces to the squared magnitude of the associated amplitude, exactly as we asserted in Schrödinger's mechanics.

39. For degenerate eigenvalues, we don't just get a single squared amplitude,
we get a sum of squared amplitudes for the different basis vector associated with that eigenvalue.
In practice, this last expression is usually the way we actually calculate the probability for degenerate eigenvalues. Find the amplitudes for each basis vector with the same eigenvalue, take their squared magnitudes, and add them up.

40. For degenerate eigenvalues, we don't just get a single squared amplitude,
we get a sum of squared amplitudes for the different basis vector associated with that eigenvalue.
In practice, this last expression is usually the way we actually calculate the probability for degenerate eigenvalues. Find the amplitudes for each basis vector with the same eigenvalue, take their squared magnitudes, and add them up.

41. For degenerate eigenvalues, we don't just get a single squared amplitude,
we get a sum of squared amplitudes for the different basis vector associated with that eigenvalue.
In practice, this last expression is usually the way we actually calculate the probability for degenerate eigenvalues. Find the amplitudes for each basis vector with the same eigenvalue, take their squared magnitudes, and add them up.

42. For degenerate eigenvalues, we don't just get a single squared amplitude,
we get a sum of squared amplitudes for the different basis vector associated with that eigenvalue.
In practice, this last expression is usually the way we actually calculate the probability for degenerate eigenvalues. Find the amplitudes for each basis vector with the same eigenvalue, take their squared magnitudes, and add them up.

43. For degenerate eigenvalues, we don't just get a single squared amplitude,
we get a sum of squared amplitudes for the different basis vector associated with that eigenvalue.
In practice, this last expression is usually the way we actually calculate the probability for degenerate eigenvalues.
Find the amplitudes for each basis vector with the same eigenvalue, take their squared magnitudes, and add them up.

44. For degenerate eigenvalues, we don't just get a single squared amplitude,
we get a sum of squared amplitudes for the different basis vector associated with that eigenvalue.
In practice, this last expression is usually the way we actually calculate the probability for degenerate eigenvalues.
Find the amplitudes for each basis vector with the same eigenvalue, take their squared magnitudes, and add them up.

45. An important part of the interpretation of the measurement process is that the value of an observable 𝐴 is really not defined unless the system is in an eigenstate associated with that observable.
If it is in such an eigenstate, let us call it then it lies entirely within the associated eigenspace 𝑆 𝑎 .
The act of a projector on such a state will be to annihilate the state if 𝑎≠𝑎′, and to leave it alone if 𝑎=𝑎′.
Thus, under such circumstances, it is clear that
𝑃(𝑎′) = 0 for 𝑎′≠𝑎
i.e. 𝑃(𝑎) = 1
Hence the probability of obtaining the eigenvalue 𝑎 associated with an eigenstate having that eigenvalue is equal to unity, which is operationally the only time that the value of the associated observable 𝐴 is well defined.

46. An important part of the interpretation of the measurement process is that the value of an observable 𝐴 is really not defined unless the system is in an eigenstate associated with that observable.
If it is in such an eigenstate, let us call it then it lies entirely within the associated eigenspace 𝑆 𝑎 .
The act of a projector on such a state will be to annihilate the state if 𝑎≠𝑎′, and to leave it alone if 𝑎=𝑎′.
Thus, under such circumstances, it is clear that
𝑃(𝑎′) = 0 for 𝑎′≠𝑎
i.e. 𝑃(𝑎) = 1
Hence the probability of obtaining the eigenvalue 𝑎 associated with an eigenstate having that eigenvalue is equal to unity, which is operationally the only time that the value of the associated observable 𝐴 is well defined.

47. An important part of the interpretation of the measurement process is that the value of an observable 𝐴 is really not defined unless the system is in an eigenstate associated with that observable.
If it is in such an eigenstate, let us call it then it lies entirely within the associated eigenspace 𝑆 𝑎 .
The act of a projector on such a state will be to annihilate the state if 𝑎≠𝑎′, and to leave it alone if 𝑎=𝑎′.
Thus, under such circumstances, it is clear that
𝑃(𝑎′) = 0 for 𝑎′≠𝑎
i.e. 𝑃(𝑎) = 1
Hence the probability of obtaining the eigenvalue 𝑎 associated with an eigenstate having that eigenvalue is equal to unity, which is operationally the only time that the value of the associated observable 𝐴 is well defined.

48. An important part of the interpretation of the measurement process is that the value of an observable 𝐴 is really not defined unless the system is in an eigenstate associated with that observable.
If it is in such an eigenstate, let us call it then it lies entirely within the associated eigenspace 𝑆 𝑎 .
The act of a projector on such a state will be to annihilate the state if 𝑎≠𝑎′, and to leave it alone if 𝑎=𝑎′.
Thus, under such circumstances, it is clear that
𝑃(𝑎′) = 0 for 𝑎′≠𝑎
i.e. 𝑃(𝑎) = 1
Hence the probability of obtaining the eigenvalue 𝑎 associated with an eigenstate having that eigenvalue is equal to unity, which is operationally the only time that the value of the associated observable 𝐴 is well defined.

49. An important part of the interpretation of the measurement process is that the value of an observable 𝐴 is really not defined unless the system is in an eigenstate associated with that observable.
If it is in such an eigenstate, let us call it then it lies entirely within the associated eigenspace 𝑆 𝑎 .
The act of a projector on such a state will be to annihilate the state if 𝑎≠𝑎′, and to leave it alone if 𝑎=𝑎′.
Thus, under such circumstances, it is clear that
𝑃(𝑎′) = 0 for 𝑎′≠𝑎
i.e. 𝑃(𝑎) = 1
Hence the probability of obtaining the eigenvalue 𝑎 associated with an eigenstate having that eigenvalue is equal to unity, which is operationally the only time that the value of the associated observable 𝐴 is well defined.

50. We now need to address the second part of the measurement postulate which describes what happens to a quantum mechanical system itself as a result of the measurement process performed upon it.
We assert that in an ideal measurement, which is one which perturbs the system as little as possible, the state of the system immediately after the measurement is one that:
is consistent with the particular eigenvalue obtained as a result of the measurement process, and
retains as much information about the state of the system immediately before the measurement as is consistent with (1).
These ideas form the basis for the following:

51. We now need to address the second part of the measurement postulate which describes what happens to a quantum mechanical system itself as a result of the measurement process performed upon it.
We assert that in an ideal measurement, which is one which perturbs the system as little as possible, the state of the system immediately after the measurement is one that:
is consistent with the particular eigenvalue obtained as a result of the measurement process, and
retains as much information about the state of the system immediately before the measurement as is consistent with (1).
These ideas form the basis for the following:

52. We now need to address the second part of the measurement postulate which describes what happens to a quantum mechanical system itself as a result of the measurement process performed upon it.
We assert that in an ideal measurement, which is one which perturbs the system as little as possible, the state of the system immediately after the measurement is one that:
is consistent with the particular eigenvalue obtained as a result of the measurement process, and
retains as much information about the state of the system immediately before the measurement as is consistent with (1).
These ideas form the basis for the following:

53. We now need to address the second part of the measurement postulate which describes what happens to a quantum mechanical system itself as a result of the measurement process performed upon it.
We assert that in an ideal measurement, which is one which perturbs the system as little as possible, the state of the system immediately after the measurement is one that:
is consistent with the particular eigenvalue obtained as a result of the measurement process, and
retains as much information about the state of the system immediately before the measurement as is consistent with (1).
These ideas form the basis for the following:

54. We now need to address the second part of the measurement postulate which describes what happens to a quantum mechanical system itself as a result of the measurement process performed upon it.
We assert that in an ideal measurement, which is one which perturbs the system as little as possible, the state of the system immediately after the measurement is one that:
is consistent with the particular eigenvalue obtained as a result of the measurement process, and
retains as much information about the state of the system immediately before the measurement as is consistent with (1).
These ideas form the basis for the following:

55. Postulate III(b) (Collapse of the State Vector)
Immediately after a measurement of an observable 𝐴 performed on a system in the state |𝜓〉 that yields the value 𝑎, the state of the system is left in the normalized projection of |𝜓〉 onto the eigenspace 𝑆 𝑎 associated with the eigenvalue measured, i.e., it is left in that part of |𝜓〉 lying within 𝑆 𝑎 .
We schematically indicate this as follows:
Thus, nature just throws away those parts of the state vector which are not consistent with the actual value obtained.

56. Postulate III(b) (Collapse of the State Vector)
Immediately after a measurement of an observable 𝐴 performed on a system in the state |𝜓〉 that yields the value 𝑎, the state of the system is left in the normalized projection of |𝜓〉 onto the eigenspace 𝑆 𝑎 associated with the eigenvalue measured, i.e., it is left in that part of |𝜓〉 lying within 𝑆 𝑎 .
We schematically indicate this as follows:
Thus, nature just throws away those parts of the state vector which are not consistent with the actual value obtained.

57. Postulate III(b) (Collapse of the State Vector)
Immediately after a measurement of an observable 𝐴 performed on a system in the state |𝜓〉 that yields the value 𝑎, the state of the system is left in the normalized projection of |𝜓〉 onto the eigenspace 𝑆 𝑎 associated with the eigenvalue measured, i.e., it is left in that part of |𝜓〉 lying within 𝑆 𝑎 .
We schematically indicate this as follows:
Thus, nature just seems to throw away those parts of the state vector which are not consistent with the actual value obtained.

58. Note that this only indicates one possible branch of the change in the system during the course of the measurement process, that which occurs when the particular eigenvalue a is obtained.
As we have seen, it is not possible to predict which of these branches will actually be followed by any single quantum mechanical system.
Thus, this change in the state vector during measurement is inherently non- deterministic.
The viewpoint usually taken is that the collapse of the state vector to the associated eigenspace occurs as the result of an unspecified interaction of the system with the (classical) measuring device used to measure the observable.

59. Note that this only indicates one possible branch of the change in the system during the course of the measurement process, that which occurs when the particular eigenvalue a is obtained.
As we have seen, it is not possible to predict which of these branches will actually be followed by any single quantum mechanical system.
Thus, this change in the state vector during measurement is inherently non- deterministic.
The viewpoint usually taken is that the collapse of the state vector to the associated eigenspace occurs as the result of an unspecified interaction of the system with the (classical) measuring device used to measure the observable.

60. Note that this only indicates one possible branch of the change in the system during the course of the measurement process, that which occurs when the particular eigenvalue a is obtained.
As we have seen, it is not possible to predict which of these branches will actually be followed by any single quantum mechanical system.
Thus, this change in the state vector during measurement is inherently non- deterministic.
The viewpoint usually taken is that the collapse of the state vector to the associated eigenspace occurs as the result of an unspecified interaction of the system with the (classical) measuring device used to measure the observable.

61. Note that this only indicates one possible branch of the change in the system during the course of the measurement process, that which occurs when the particular eigenvalue a is obtained.
As we have seen, it is not possible to predict which of these branches will actually be followed by any single quantum mechanical system.
Thus, this change in the state vector during measurement is inherently non- deterministic.
The viewpoint usually taken is that the collapse of the state vector to the associated eigenspace occurs as the result of an unspecified interaction of the system with the (classical) measuring device used to measure the observable.

62. Note that this only indicates one possible branch of the change in the system during the course of the measurement process, that which occurs when the particular eigenvalue a is obtained.
As we have seen, it is not possible to predict which of these branches will actually be followed by any single quantum mechanical system.
Thus, this change in the state vector during measurement is inherently non- deterministic.
The viewpoint usually taken is that the collapse of the state vector to the associated eigenspace occurs as the result of an unspecified interaction of the system with the (classical) measuring device used to measure the observable.

63. Extension to Continuous Eigenvalues As with Schrödinger's mechanics, we have initially stated this postulate in a form which assumes that the spectrum of the observable of interest is discrete.
We now discuss how the 3rd postulate needs to be modified for the case of observables with a continuous spectrum of eigenvalues {𝛼}.
In this circumstance, A still has an ONB of eigenvectors {|𝛼,𝜏〉}
which satisfy the completeness relation and the Dirac orthonormality condition
and in terms of which we can expand the state

64. Extension to Continuous Eigenvalues As with Schrödinger's mechanics, we have initially stated this postulate in a form which assumes that the spectrum of the observable of interest is discrete.
We now discuss how the 3rd postulate needs to be modified for the case of observables with a continuous spectrum of eigenvalues {𝛼}.
In this circumstance, A still has an ONB of eigenvectors {|𝛼,𝜏〉}
which satisfy the completeness relation and the Dirac orthonormality condition
and in terms of which we can expand the state

65. Extension to Continuous Eigenvalues As with Schrödinger's mechanics, we have initially stated this postulate in a form which assumes that the spectrum of the observable of interest is discrete.
We now discuss how the 3rd postulate needs to be modified for the case of observables with a continuous spectrum of eigenvalues {𝛼}.
In this circumstance, 𝐴 still has an ONB of eigenvectors {|𝛼,𝜏〉}
which satisfy the completeness relation and the Dirac orthonormality condition
and in terms of which we can expand the state

66. Extension to Continuous Eigenvalues As with Schrödinger's mechanics, we have initially stated this postulate in a form which assumes that the spectrum of the observable of interest is discrete.
We now discuss how the 3rd postulate needs to be modified for the case of observables with a continuous spectrum of eigenvalues {𝛼}.
In this circumstance, 𝐴 still has an ONB of eigenvectors {|𝛼,𝜏〉} which satisfy the the Dirac orthonormality condition and the completeness relation
and in terms of which we can expand the state

67. Extension to Continuous Eigenvalues As with Schrödinger's mechanics, we have initially stated this postulate in a form which assumes that the spectrum of the observable of interest is discrete.
We now discuss how the 3rd postulate needs to be modified for the case of observables with a continuous spectrum of eigenvalues {𝛼}.
In this circumstance, 𝐴 still has an ONB of eigenvectors {|𝛼,𝜏〉} which satisfy the the Dirac orthonormality condition and the completeness relation
and in terms of which we can expand the state

68. Extension to Continuous Eigenvalues As with Schrödinger's mechanics, we have initially stated this postulate in a form which assumes that the spectrum of the observable of interest is discrete.
We now discuss how the 3rd postulate needs to be modified for the case of observables with a continuous spectrum of eigenvalues {𝛼}.
In this circumstance, 𝐴 still has an ONB of eigenvectors {|𝛼,𝜏〉} which satisfy the the Dirac orthonormality condition and the completeness relation
and in terms of which we can expand the state

69. Extension to Continuous Eigenvalues As with Schrödinger's mechanics, we have initially stated this postulate in a form which assumes that the spectrum of the observable of interest is discrete.
We now discuss how the 3rd postulate needs to be modified for the case of observables with a continuous spectrum of eigenvalues {𝛼}.
In this circumstance, 𝐴 still has an ONB of eigenvectors {|𝛼,𝜏〉} which satisfy the the Dirac orthonormality condition and the completeness relation
and in terms of which we can expand the state

70. Extension to Continuous Eigenvalues As with Schrödinger's mechanics, we have initially stated this postulate in a form which assumes that the spectrum of the observable of interest is discrete.
We now discuss how the 3rd postulate needs to be modified for the case of observables with a continuous spectrum of eigenvalues {𝛼}.
In this circumstance, 𝐴 still has an ONB of eigenvectors {|𝛼,𝜏〉} which satisfy the the Dirac orthonormality condition and the completeness relation
and in terms of which we can expand the state

71. where the expansion coefficients are just given by the inner products
Now, the completeness relation can be written in the form
where, for fixed 𝛼,
is the sum of the orthogonal projector densities onto the basis vectors that span the eigenspace 𝑆 𝛼 of 𝐴 with eigenvalue 𝛼. It is thus the total projector density associated with that eigenvalue.

72. where the expansion coefficients are just given by the inner products
Now, the completeness relation can be written in the form
where, for fixed 𝛼,
is the sum of the orthogonal projector densities onto the basis vectors that span the eigenspace 𝑆 𝛼 of 𝐴 with eigenvalue 𝛼. It is thus the total projector density associated with that eigenvalue.

73. where the expansion coefficients are just given by the inner products
Now, the completeness relation can be written in the form
where, for fixed 𝛼,
is the sum of the orthogonal projector densities onto the basis vectors that span the eigenspace 𝑆 𝛼 of 𝐴 with eigenvalue 𝛼. It is thus the total projector density associated with that eigenvalue.

74. where the expansion coefficients are just given by the inner products
Now, the completeness relation can be written in the form
where, for fixed 𝛼,
is the sum of the orthogonal projector densities onto the basis vectors that span the eigenspace 𝑆 𝛼 of 𝐴 with eigenvalue 𝛼.
It is thus the total projector density associated with that eigenvalue.

75. where the expansion coefficients are just given by the inner products
Now, the completeness relation can be written in the form
where, for fixed 𝛼,
is the sum of the orthogonal projector densities onto the basis vectors that span the eigenspace 𝑆 𝛼 of 𝐴 with eigenvalue 𝛼.
It is thus the total projector density associated with that eigenvalue.

76. We can therefore write
where
is just the part of the state |𝜓〉 lying in the eigenspace 𝑆 𝛼 .
Then, in describing measurements of an observable 𝐴 with a continuous spectrum, rather than discuss the probability of obtaining a particular eigenvalue (which is zero), we talk of the probability density 𝜌(𝛼).

77. We can therefore write
where
is just the part of the state |𝜓〉 lying in the eigenspace 𝑆 𝛼 .
Then, in describing measurements of an observable 𝐴 with a continuous spectrum, rather than discuss the probability of obtaining a particular eigenvalue (which is zero), we talk of the probability density 𝜌(𝛼).

78. We can therefore write
where
is just the part of the state |𝜓〉 lying in the eigenspace 𝑆 𝛼 .
Then, in describing measurements of an observable 𝐴 with a continuous spectrum, rather than discuss the probability of obtaining a particular eigenvalue (which is zero), we talk of the probability density 𝜌(𝛼).

79. We can therefore write
where
is just the part of the state |𝜓〉 lying in the eigenspace 𝑆 𝛼 .
Then, in describing measurements of an observable 𝐴 with a continuous spectrum, rather than discuss the probability of obtaining a particular eigenvalue (which is zero), we talk of the probability density 𝜌(𝛼).

80. We can therefore write
where
is just the part of the state |𝜓〉 lying in the eigenspace 𝑆 𝛼 .
Then, in describing measurements of an observable 𝐴 with a continuous spectrum, rather than discuss the probability of obtaining a particular eigenvalue (which is zero), we talk of the probability density 𝜌(𝛼).

81. We can therefore write
where
is just the part of the state |𝜓〉 lying in the eigenspace 𝑆 𝛼 .
Then, in describing measurements of an observable 𝐴 with a continuous spectrum, rather than discuss the probability of obtaining a particular eigenvalue (which is zero), we talk of the probability density 𝜌(𝛼).

82. We thus have the following modification:
The probability density 𝜌(𝛼) that a measurement of 𝐴 on the state |𝜓〉 will yield one of the continuous eigenvalues 𝛼 of 𝐴 is given by the expression
where is the projector density onto the eigenspace 𝑆 𝛼 associated with that eigenvalue.

83. We thus have the following modification:
The probability density 𝜌(𝛼) that a measurement of 𝐴 on the state |𝜓〉 will yield one of the continuous eigenvalues 𝛼 of 𝐴 is given by the expectation value
of the projector density taken with respect to the state |𝜓〉.

84. Note that, if the eigenvalue 𝛼 is nondegenerate, so that there is only one linearly independent basis vector |𝛼〉, then
In this limit,
Thus, the probability density in this case reduces to the squared magnitude of the associated wave function, exactly as we saw in Schrödinger's mechanics.
Thus, for a single particle, we have
and similarly

85. Note that, if the eigenvalue 𝛼 is nondegenerate, so that there is only one linearly independent basis vector |𝛼〉, then
In this limit,
Thus, the probability density in this case reduces to the squared magnitude of the associated wave function, exactly as we saw in Schrödinger's mechanics.
Thus, for a single particle, we have
and similarly

86. Note that, if the eigenvalue 𝛼 is nondegenerate, so that there is only one linearly independent basis vector |𝛼〉, then
In this limit,
Thus, the probability density in this case reduces to the squared magnitude of the associated wave function, exactly as we saw in Schrödinger's mechanics.
Thus, for a single particle, we have
and similarly

87. Note that, if the eigenvalue 𝛼 is nondegenerate, so that there is only one linearly independent basis vector |𝛼〉, then
In this limit,
Thus, the probability density in this case reduces to the squared magnitude of the associated wave function, exactly as we saw in Schrödinger's mechanics.
Thus, for a single particle, we have
and similarly

88. Note that, if the eigenvalue 𝛼 is nondegenerate, so that there is only one linearly independent basis vector |𝛼〉, then
In this limit,
Thus, the probability density in this case reduces to the squared magnitude of the associated wave function, exactly as we saw in Schrödinger's mechanics.
Thus, for a single particle, we have
and similarly

89. For degenerate eigenvalues, we don't just get a single squared amplitude
we get a sum of squared amplitudes for the different basis vector associated with that eigenvalue.
Note in the expressions that we have given above, the index 𝜏, which we have written as though it were discrete, can sometimes be continuous, if the associated eigenvalue is infinitely degenerate. In that case, all of the expressions above involving a sum over 𝜏 can be modified by re-writing in terms of an integral

90. For degenerate eigenvalues, we don't just get a single squared amplitude
we get a sum of squared amplitudes for the different basis vector associated with that eigenvalue.
Note in the expressions that we have given above, the index 𝜏, which we have written as though it were discrete, can sometimes be continuous, if the associated eigenvalue is infinitely degenerate. In that case, all of the expressions above involving a sum over 𝜏 can be modified by re-writing in terms of an integral

91. For degenerate eigenvalues, we don't just get a single squared amplitude
we get a sum of squared amplitudes for the different basis vector associated with that eigenvalue.
Note in the expressions that we have given above, the index 𝜏, which we have written as though it were discrete, can sometimes be continuous, if the associated eigenvalue is infinitely degenerate. In that case, all of the expressions above involving a sum over 𝜏 can be modified by re-writing in terms of an integral

92. For degenerate eigenvalues, we don't just get a single squared amplitude
we get a sum of squared amplitudes for the different basis vector associated with that eigenvalue.
Note in the expressions that we have given above, the index 𝜏, which we have written as though it were discrete, can sometimes be continuous, if the associated eigenvalue is infinitely degenerate. In that case, all of the expressions above involving a sum over 𝜏 can be modified by re-writing in terms of an integral

93. For degenerate eigenvalues, we don't just get a single squared amplitude
we get a sum of squared amplitudes for the different basis vector associated with that eigenvalue.
Note in the expressions that we have given above, the index 𝜏, which we have written as though it were discrete, can sometimes be continuous, if the associated eigenvalue is infinitely degenerate. In that case, all of the expressions above involving a sum over 𝜏 can be modified by re-writing in terms of an integral

94. For degenerate eigenvalues, we don't just get a single squared amplitude
we get a sum of squared amplitudes for the different basis vector associated with that eigenvalue.
Note in the expressions that we have given above, the index 𝜏, which we have written as though it were discrete, can sometimes be continuous, if the associated eigenvalue is infinitely degenerate. In that case, all of the expressions above involving a sum over 𝜏 can be modified by re-writing in terms of an integral

95. For degenerate eigenvalues, we don't just get a single squared amplitude
we get a sum of squared amplitudes for the different basis vector associated with that eigenvalue.
Note in the expressions that we have given above, the index 𝜏, which we have written as though it were discrete, can sometimes be continuous, if the associated eigenvalue is infinitely degenerate. In that case, all of the expressions above involving a sum over 𝜏 can be modified by re-writing in terms of an integral

96. In this lecture, we stated the 3rd postulate in a form that takes into account the possible degeneracy of the eigenvalues of an observable being measured.
In any measurement of an observable A, the only values that can be obtained is one of the eigenvalues of that observable. The 3rd postulate gives an expression for the probability, or probability density to obtain an eigenvalue in the discrete or continuous part of the spectrum.
When a particular eigenvalue is obtained in a measurement, the state vector collapses onto the eigenspace associated with that eigenvalue. The parts associated with different eigenvalues are simply removed.
In the next lecture, we consider some examples, and discuss implications of the third postulate.

97. In this lecture, we stated the 3rd postulate in a form that takes into account the possible degeneracy of the eigenvalues of an observable being measured.
In any measurement of an observable 𝐴, the only values that can be obtained are the eigenvalues of that observable. The 3rd postulate gives an expression for the probability, or probability density to obtain an eigenvalue in the discrete or continuous part of the spectrum.
When a particular eigenvalue is obtained in a measurement, the state vector collapses onto the eigenspace associated with that eigenvalue. The parts associated with different eigenvalues are simply removed.
In the next lecture, we consider some examples, and discuss implications of the third postulate.

98. In this lecture, we stated the 3rd postulate in a form that takes into account the possible degeneracy of the eigenvalues of an observable being measured.
In any measurement of an observable 𝐴, the only values that can be obtained are the eigenvalues of that observable. The 3rd postulate gives an expression for the probability, or probability density to obtain an eigenvalue in the discrete or continuous part of the spectrum.
When a particular eigenvalue is obtained in a measurement, the state vector collapses onto the eigenspace associated with that eigenvalue. The parts associated with different eigenvalues are simply removed.
In the next lecture, we consider some examples, and discuss implications of the third postulate.

99. In this lecture, we stated the 3rd postulate in a form that takes into account the possible degeneracy of the eigenvalues of an observable being measured.
In any measurement of an observable 𝐴, the only values that can be obtained are the eigenvalues of that observable. The 3rd postulate gives an expression for the probability, or probability density to obtain an eigenvalue in the discrete or continuous part of the spectrum.
When a particular eigenvalue is obtained in a measurement, the state vector collapses onto the eigenspace associated with that eigenvalue. The parts associated with different eigenvalues are simply removed.
In the next lecture, we consider some examples, and discuss implications of the third postulate.