I
The Schr¨ odinger and Heisenberg Pictures
The standard introductory formalism of quantum mechanics is that of Schr¨ odinger, in which the dynamical variables (such as x and p ˆ = −i∇) are independent of time, while the wavefunction ψ changes with time according to the
general equation
∂ψ(x, t)
ˆ
Hψ(x, t) = i
(I.1)
∂t
where H ˆ is the Hamiltonian. Matrix elements of operators A ˆ depending on
x, p ˆ . . . then have the form
∫
<φ|A ˆ |ψ> = φ
∗ (x, t) ˆ
x
(I.2)
Aψ(x, t) d
3
and will, in general, depend on time via the time dependences of φ and ψ.
Although used almost universally in introductory courses on quantum mechanics, this formulation is not the only possible one, nor is it always the
most convenient.
We may, for example, wish to bring out similarities (and differences) between the general dynamical frameworks of quantum and classical mechanics.
The formulation here does not seem to be well adapted to this purpose, since
in the classical case the dynamical variables depend on time (x(t), p(t) . . .)
and obey equations of motion, while the quantum variables A ˆ are timeindependent and the ‘equation of motion’ (I.1) is for the wavefunction ψ,
which has no classical counterpart. In quantum mechanics, however, it is
always possible to make unitary transformations of the state vector or wavefunctions. We can make use of this possibility to obtain an alternative formulation of quantum mechanics, which is in some ways closer to the spirit of
classical mechanics, as follows.
Equation (I.1) can be formally solved to give
−i ˆ
ψ(x, t) = e
Ht ψ(x, 0)
(I.3)
where the exponential (of an operator!) can be defined by the corresponding
power series, for example:
Ht = 1 − i ˆ
1 (−i ˆ
e
−i ˆ
Ht +
Ht)
2 + · · · .
(I.4)
2!
It is simple to check that (I.3) as defined by (I.4), does satisfy (I.1) and that
405
The Schr¨ odinger and Heisenberg Pictures
The standard introductory formalism of quantum mechanics is that of Schr¨ odinger, in which the dynamical variables (such as x and p ˆ = −i∇) are independent of time, while the wavefunction ψ changes with time according to the
general equation
∂ψ(x, t)
ˆ
Hψ(x, t) = i
(I.1)
∂t
where H ˆ is the Hamiltonian. Matrix elements of operators A ˆ depending on
x, p ˆ . . . then have the form
∫
<φ|A ˆ |ψ> = φ
∗ (x, t) ˆ
x
(I.2)
Aψ(x, t) d
3
and will, in general, depend on time via the time dependences of φ and ψ.
Although used almost universally in introductory courses on quantum mechanics, this formulation is not the only possible one, nor is it always the
most convenient.
We may, for example, wish to bring out similarities (and differences) between the general dynamical frameworks of quantum and classical mechanics.
The formulation here does not seem to be well adapted to this purpose, since
in the classical case the dynamical variables depend on time (x(t), p(t) . . .)
and obey equations of motion, while the quantum variables A ˆ are timeindependent and the ‘equation of motion’ (I.1) is for the wavefunction ψ,
which has no classical counterpart. In quantum mechanics, however, it is
always possible to make unitary transformations of the state vector or wavefunctions. We can make use of this possibility to obtain an alternative formulation of quantum mechanics, which is in some ways closer to the spirit of
classical mechanics, as follows.
Equation (I.1) can be formally solved to give
−i ˆ
ψ(x, t) = e
Ht ψ(x, 0)
(I.3)
where the exponential (of an operator!) can be defined by the corresponding
power series, for example:
Ht = 1 − i ˆ
1 (−i ˆ
e
−i ˆ
Ht +
Ht)
2 + · · · .
(I.4)
2!
It is simple to check that (I.3) as defined by (I.4), does satisfy (I.1) and that
405