1. Quantum One

3. Using a Complete Set of Commuting Observables
Preparation of States
Using a Complete Set of Commuting Observables

4. In the last lecture, we explored some consequences of the 3rd postulate, dealing with the measurement of quantum mechanical systems.
We saw that the average value obtained in a large number of measurements of an observable 𝐴 on a set of identically prepared states can be simply expressed in terms of the expectation value of 𝐴 taken with respect to the state on which the measurements are made.
We also introduced a notion of the statistical width or uncertainty associated with quantum mechanical measurement, introducing the RMS uncertainty, which can be computed from the mean value 𝐴 and the mean value of 𝐴 2 .
We also proved the uncertainty relation for two arbitrary observables, which indicates the degree to which they can be simultaneously known, and which reduces to Heisenberg’s uncertainty relation for position and momentum.

5. Clearly, in order to actually test the third postulate, it is necessary to generate an ensemble of quantum mechanical systems that are all in the same physical state, since the content of that postulate involves statistical predictions that apply collectively to such an ensemble.
For internal consistency, therefore, it is necessary to specify how such an ensemble could, in principle, be generated, assuming that the postulate is valid.
To do this, assume that one starts with an arbitrary ensemble of physically similar quantum systems, in which the individual elements are in arbitrary initial states, {|𝜓〉} which may or may not be known at the outset.
According to the postulates, it is then possible to measure compatible observables, or commuting observables, either simultaneously or sequentially over a brief period of time.

6. Clearly, in order to actually test the third postulate, it is necessary to generate an ensemble of quantum mechanical systems that are all in the same physical state, since the content of that postulate involves statistical predictions that apply collectively to such an ensemble.
For internal consistency, therefore, it is necessary to specify how such an ensemble could, in principle, be generated, assuming that the postulate is valid.
To do this, assume that one starts with an arbitrary ensemble of physically similar quantum systems, in which the individual elements are in arbitrary initial states, {|𝜓〉} which may or may not be known at the outset.
According to the postulates, it is then possible to measure compatible observables, or commuting observables, either simultaneously or sequentially over a brief period of time.

7. Clearly, in order to actually test the third postulate, it is necessary to generate an ensemble of quantum mechanical systems that are all in the same physical state, since the content of that postulate involves statistical predictions that apply collectively to such an ensemble.
For internal consistency, therefore, it is necessary to specify how such an ensemble could, in principle, be generated, assuming that the postulate is valid.
To do this, assume that one starts with an arbitrary ensemble of physically similar quantum systems, in which the individual elements are in arbitrary initial states, {|𝜓〉} which may or may not be known at the outset.
According to the postulates, it is then possible to measure compatible observables, or commuting observables, either simultaneously or sequentially over a brief period of time.

8. Clearly, in order to actually test the third postulate, it is necessary to generate an ensemble of quantum mechanical systems that are all in the same physical state, since the content of that postulate involves statistical predictions that apply collectively to such an ensemble.
For internal consistency, therefore, it is necessary to specify how such an ensemble could, in principle, be generated, assuming that the postulate is valid.
To do this, assume that one starts with an arbitrary ensemble of physically similar quantum systems, in which the individual elements are in arbitrary initial states, {|𝜓〉} which may or may not be known at the outset.
According to the postulates, it is then possible to measure compatible observables, or commuting observables, either simultaneously or sequentially over a brief period of time on each of the members of this ensemble.

9. Suppose, therefore, the each member of the ensemble is subjected to the measurement of a complete set of commuting observables {𝐴,𝐵,𝐶}, say.
For such a set, there exists, by assumption, a complete ONB of common eigenstates {|𝑎,𝑏,𝑐〉}, each element of which is uniquely labeled by the corresponding set of eigenvalues 𝑎,𝑏,𝑐.
For a member of our ensemble initially in a quantum state
a measurement of these three compatible observables in a very short interval of time, we will see the subsequent reduction of the state vector onto just one of these basis vectors.

10. Suppose, therefore, the each member of the ensemble is subjected to the measurement of a complete set of commuting observables {𝐴,𝐵,𝐶}, say.
For such a set, there exists, by assumption, a complete ONB of common eigenstates {|𝑎,𝑏,𝑐〉}, each element of which is uniquely labeled by the corresponding set of eigenvalues 𝑎,𝑏,𝑐.
For a member of our ensemble initially in a quantum state
a measurement of these three compatible observables in a very short interval of time, we will see the subsequent reduction of the state vector onto just one of these basis vectors.

11. Suppose, therefore, the each member of the ensemble is subjected to the measurement of a complete set of commuting observables {𝐴,𝐵,𝐶}, say.
For such a set, there exists, by assumption, a complete ONB of common eigenstates {|𝑎,𝑏,𝑐〉}, each element of which is uniquely labeled by the corresponding set of eigenvalues 𝑎,𝑏,𝑐.
For a member of our ensemble initially in a quantum state
a measurement of these three compatible observables in a very short interval of time, we will see the subsequent reduction of the state vector onto just one of these basis vectors.

12. Suppose, therefore, the each member of the ensemble is subjected to the measurement of a complete set of commuting observables {𝐴,𝐵,𝐶}, say.
For such a set, there exists, by assumption, a complete ONB of common eigenstates {|𝑎,𝑏,𝑐〉}, each element of which is uniquely labeled by the corresponding set of eigenvalues 𝑎,𝑏,𝑐.
For a member of our ensemble initially in a quantum state
a measurement of these three compatible observables in a very short interval of time will bring about the collapse of the state vector onto just one of these basis vectors.

13. This is schematically indicated in the following diagram
where we have performed the required re-normalization in the last step, in terms of the phase factor

14. This is schematically indicated in the following diagram
where we have performed the required re-normalization in the last step, in terms of the phase factor

15. This is schematically indicated in the following diagram
where we have performed the required re-normalization in the last step, in terms of the phase factor

16. This is schematically indicated in the following diagram
where we have performed the required re-normalization in the last step, in terms of the phase factor

17. Thus, each member of the ensemble is left, up to a phase factor, in a well-defined physical state, which is now known as a result of the measurement process.
Thus, after performing such a complete series of measurements on an ensemble of arbitrary initial state vectors, we can extract those which end up, within a phase factor, in a particular quantum state |𝑎,𝑏,𝑐〉 to produce a sub-ensemble of systems upon which to perform further experiments, and thereby test the statistical predictions of the postulate.
Note that predictions of the postulate, which are stated in terms of expectation values, are insensitive to such a phase factor. That is, if we consider 2 states
and
then while

18. Thus, each member of the ensemble is left, up to a phase factor, in a well-defined physical state, which is now know as a result of the measurement process.
Thus, after performing such a complete series of measurements on an ensemble of arbitrary initial state vectors, we can extract those which end up, to within a phase factor, in a particular quantum state |𝑎,𝑏,𝑐〉 to produce a sub-ensemble of systems upon which to perform further experiments, and thereby test the statistical predictions of the postulate.
Note that predictions of the postulate, which are stated in terms of expectation values, are insensitive to such a phase factor. That is, if we consider 2 states
and
then while

19. Thus, each member of the ensemble is left, up to a phase factor, in a well-defined physical state, which is now know as a result of the measurement process.
Thus, after performing such a complete series of measurements on an ensemble of arbitrary initial state vectors, we can extract those which end up, within a phase factor, in a particular quantum state |𝑎,𝑏,𝑐〉 to produce a sub-ensemble of systems upon which to perform further experiments, and thereby test the statistical predictions of the postulate.
Note that predictions of the postulate, which are stated in terms of expectation values, are insensitive to such a phase factor. That is, if we consider 2 states
and
then while

20. Thus, each member of the ensemble is left, up to a phase factor, in a well-defined physical state, which is now known as a result of the measurement process.
Thus, after performing such a complete series of measurements on an ensemble of arbitrary initial state vectors, we can extract those which end up, within a phase factor, in a particular quantum state |𝑎,𝑏,𝑐〉 to produce a sub-ensemble of systems upon which to perform further experiments, and thereby test the statistical predictions of the postulate.
Note that predictions of the postulate, which are stated in terms of expectation values, are insensitive to such a phase factor. That is, if we consider 2 states
and
then while

21. Thus, each member of the ensemble is left, up to a phase factor, in a well-defined physical state, which is now known as a result of the measurement process.
Thus, after performing such a complete series of measurements on an ensemble of arbitrary initial state vectors, we can extract those which end up, within a phase factor, in a particular quantum state |𝑎,𝑏,𝑐〉 to produce a sub-ensemble of systems upon which to perform further experiments, and thereby test the statistical predictions of the postulate.
Note that predictions of the postulate, which are stated in terms of expectation values, are insensitive to such a phase factor. That is, if we consider 2 states
and
then while