10.1 Quantum Mechanics
147
In quantum mechanics all physical properties of states are described by hermitian
operators, denoted by ˆ
O, and labeled with a caret. Measuring the property ˆ
O of a
state | f involves probing the result of ˆ
O| f with a second state g|, which involves
calculating the matrix-element g| ˆ
O| f . This procedure describes calculating the
overlap of ˆ
O’s action on | f with g|. The probability of finding ˆ
O| f in g| is
then given by the squared absolute value of the matrix element ||g| ˆ
O| f |
2
. We
can explicitely calculate these matrix elements by inserting the identity from (10.3)
between the states and the operator
g| ˆ
O| f =
dx
dxg|x
x
| ˆ
O|xx| f
=
dx
dx g(x
)
∗ O(x
, x) f (x)
(10.6)
and O(x
, x) = =x
| ˆ
O|x is the operator ˆ
O expressed in the basis of position vectors.
We could have equally well used the momentum base vectors from (10.5) and had
obtained the states | f and g| and the operator ˆ
O expressed in “momentum space.”
All operators representing physically observable quantities are described by hermitian operators. Their action on a vector or its dual vector gives the same result. To
see this, we first define the hermitian conjugate operator to operator ˆ
O by a dagger
ˆ
O
† and that it operates on the dual bra-state on its left-hand side via
f | ˆ
O
†
|g = =g| ˆ
O| f
∗
,
(10.7)
where, as before, the asterisk denotes the complex conjugate.
The momentum operator ˆ
p has the form ˆ
p = −i∂/∂ x and it is easy to see that
x| ˆ
p|k = −i
∂
∂ x
x|k = −i
∂
∂ x
e
ikx
= ke
ikx
= px|k .
(10.8)
Applying ˆ
p to a state with k = p/ thus produces a real number p = k and represents a measurement of the momentum of the state |k.
In the Schrödinger formulation of quantum mechanics, particles are described by
waves (x, t) ∝ e
ikx−iωt
, characterized by a wave vector k = p/ and a frequency
ω, which is related to the energy E of the wave by Planck’s relation E = ω.
We see that we can determine the energy E by a partial derivative with respect
to the time t, such that i∂∂(x, t)/∂t = E(x, t). Furthermore, we know from
classical mechanics that the (kinetic) energy of a freely moving particle is related
to the momentum p by E = p
2
/2m with the mass m of the particle. Following
Schrödinger’s daring step to interpret this relation as an operator equation, we write
E = ˆ
p
2
/2m = −(
2
/2m)(∂/∂ x)
2
. Inserting in the equation with the time-derivative
we obtain the Schrödinger equation for a free particle
i
∂
∂t
(x, t) = −
2
2m
∂
2
∂ x 2 (x, t) = ˆ
H (x, t) .
(10.9)
147
In quantum mechanics all physical properties of states are described by hermitian
operators, denoted by ˆ
O, and labeled with a caret. Measuring the property ˆ
O of a
state | f involves probing the result of ˆ
O| f with a second state g|, which involves
calculating the matrix-element g| ˆ
O| f . This procedure describes calculating the
overlap of ˆ
O’s action on | f with g|. The probability of finding ˆ
O| f in g| is
then given by the squared absolute value of the matrix element ||g| ˆ
O| f |
2
. We
can explicitely calculate these matrix elements by inserting the identity from (10.3)
between the states and the operator
g| ˆ
O| f =
dx
dxg|x
x
| ˆ
O|xx| f
=
dx
dx g(x
)
∗ O(x
, x) f (x)
(10.6)
and O(x
, x) = =x
| ˆ
O|x is the operator ˆ
O expressed in the basis of position vectors.
We could have equally well used the momentum base vectors from (10.5) and had
obtained the states | f and g| and the operator ˆ
O expressed in “momentum space.”
All operators representing physically observable quantities are described by hermitian operators. Their action on a vector or its dual vector gives the same result. To
see this, we first define the hermitian conjugate operator to operator ˆ
O by a dagger
ˆ
O
† and that it operates on the dual bra-state on its left-hand side via
f | ˆ
O
†
|g = =g| ˆ
O| f
∗
,
(10.7)
where, as before, the asterisk denotes the complex conjugate.
The momentum operator ˆ
p has the form ˆ
p = −i∂/∂ x and it is easy to see that
x| ˆ
p|k = −i
∂
∂ x
x|k = −i
∂
∂ x
e
ikx
= ke
ikx
= px|k .
(10.8)
Applying ˆ
p to a state with k = p/ thus produces a real number p = k and represents a measurement of the momentum of the state |k.
In the Schrödinger formulation of quantum mechanics, particles are described by
waves (x, t) ∝ e
ikx−iωt
, characterized by a wave vector k = p/ and a frequency
ω, which is related to the energy E of the wave by Planck’s relation E = ω.
We see that we can determine the energy E by a partial derivative with respect
to the time t, such that i∂∂(x, t)/∂t = E(x, t). Furthermore, we know from
classical mechanics that the (kinetic) energy of a freely moving particle is related
to the momentum p by E = p
2
/2m with the mass m of the particle. Following
Schrödinger’s daring step to interpret this relation as an operator equation, we write
E = ˆ
p
2
/2m = −(
2
/2m)(∂/∂ x)
2
. Inserting in the equation with the time-derivative
we obtain the Schrödinger equation for a free particle
i
∂
∂t
(x, t) = −
2
2m
∂
2
∂ x 2 (x, t) = ˆ
H (x, t) .
(10.9)