11.2 Quantum Behavior of Matter
517
electron beam with lots of electrons in one quantum state. However, in some
metals and semiconductors, electron-lattice interactions allow electrons to have
a very weak attraction, and can pair up at sufficiently low temperatures. These
pairs act as bosons, and make the material a superconductor.
7. Observation may disturb the state of a system. Each observation of a system
may (depending on the nature of the measurement) ‘collapse’ the wave function
into one of the possible observable states of the system. However, for quantum
theory consistency, the act of measurement is an event which opens a previously
isolated system to the measuring instrument (or the environment), possibly
making entropy flow into the original system. The wave function for both
the observed system and measuring system becomes entangled 3 when the
measurement occurs. There may be a dramatic change in the wave function
when the interaction begins, but there is no ‘wave-function collapse’ of the
combined system that includes the measuring device.
8. In quantum theory, for each observable, such as energy, linear momentum,
angular momentum, and position, there is an associated operator
A which acts
on states of the observed system. 4 The average of repeated measurement
of an observable
A of a prepared system is predicted by quantum theory to
be A =
∗
AAdx. Each observable
A has a spectrum of possible values
which can be found by a measurement of
A, called the eigenvalues {a i } of A,
i = 1, 2, · · · . Solutions to the relation
Aψ a i = a i ψ a i determine the possible
‘eigenstates’, ψ a i , and ‘eigenvalues’ a i . Since
A is an observable, the a i must
be real numbers. The eigenstates for distinct eigenvalues will be orthogonal:
ψ ∗
a i
ψ a j dx = δ ij . The eigenstates ψ a i for all possible eigenvalues a i can be
shown to form a complete set, so that any state can be written as a linear
superposition of such states: =
i c i ψ a i . After a measurement of
A which
gives an eigenvalue a i , the system wave ‘collapses’ to ψ a i . 5
9. If the relation
A
B =
B
A (i.e.
A and
B commute) is true, then the observables
A and
B can be measured in either order without disturbing the state formed by
the first measurement. The system is left in an eigenstate of both
A and
B.
10. If the observables
A and
B do not commute, then the measurement of one
followed by the second necessarily changes the state of the system. Define the
uncertainty of
A after a set of measurements of
A by the root-mean-square
3 An entangled state is a multiparticle quantum wave which cannot be factored into component
particle waves.
4 A ‘hat’ on a symbol indicates that the symbol represents an operator, rather than a number.
Operators act on quantum states, changing them to a different quantum state.
5 This phrase ‘collapse of the wave function’ has been used since Bohr’s original thoughts on
quantum theory. But as we have noted the phrase cannot be taken too literally. A consistent
quantum theory requires the following: If one includes the observer as well as the observed in
an isolated system, then the interaction of the observer with the observed in the process of a
measurement may change the state of the observed part to be predominantly in one eigenstate
of an observable, but in order for the sum of all probabilities to remain one, all the other possible
states of the observed system must still be present, now ‘entangled’ with the observer’s states.
517
electron beam with lots of electrons in one quantum state. However, in some
metals and semiconductors, electron-lattice interactions allow electrons to have
a very weak attraction, and can pair up at sufficiently low temperatures. These
pairs act as bosons, and make the material a superconductor.
7. Observation may disturb the state of a system. Each observation of a system
may (depending on the nature of the measurement) ‘collapse’ the wave function
into one of the possible observable states of the system. However, for quantum
theory consistency, the act of measurement is an event which opens a previously
isolated system to the measuring instrument (or the environment), possibly
making entropy flow into the original system. The wave function for both
the observed system and measuring system becomes entangled 3 when the
measurement occurs. There may be a dramatic change in the wave function
when the interaction begins, but there is no ‘wave-function collapse’ of the
combined system that includes the measuring device.
8. In quantum theory, for each observable, such as energy, linear momentum,
angular momentum, and position, there is an associated operator
A which acts
on states of the observed system. 4 The average of repeated measurement
of an observable
A of a prepared system is predicted by quantum theory to
be A =
∗
AAdx. Each observable
A has a spectrum of possible values
which can be found by a measurement of
A, called the eigenvalues {a i } of A,
i = 1, 2, · · · . Solutions to the relation
Aψ a i = a i ψ a i determine the possible
‘eigenstates’, ψ a i , and ‘eigenvalues’ a i . Since
A is an observable, the a i must
be real numbers. The eigenstates for distinct eigenvalues will be orthogonal:
ψ ∗
a i
ψ a j dx = δ ij . The eigenstates ψ a i for all possible eigenvalues a i can be
shown to form a complete set, so that any state can be written as a linear
superposition of such states: =
i c i ψ a i . After a measurement of
A which
gives an eigenvalue a i , the system wave ‘collapses’ to ψ a i . 5
9. If the relation
A
B =
B
A (i.e.
A and
B commute) is true, then the observables
A and
B can be measured in either order without disturbing the state formed by
the first measurement. The system is left in an eigenstate of both
A and
B.
10. If the observables
A and
B do not commute, then the measurement of one
followed by the second necessarily changes the state of the system. Define the
uncertainty of
A after a set of measurements of
A by the root-mean-square
3 An entangled state is a multiparticle quantum wave which cannot be factored into component
particle waves.
4 A ‘hat’ on a symbol indicates that the symbol represents an operator, rather than a number.
Operators act on quantum states, changing them to a different quantum state.
5 This phrase ‘collapse of the wave function’ has been used since Bohr’s original thoughts on
quantum theory. But as we have noted the phrase cannot be taken too literally. A consistent
quantum theory requires the following: If one includes the observer as well as the observed in
an isolated system, then the interaction of the observer with the observed in the process of a
measurement may change the state of the observed part to be predominantly in one eigenstate
of an observable, but in order for the sum of all probabilities to remain one, all the other possible
states of the observed system must still be present, now ‘entangled’ with the observer’s states.