6-6 The Postulate for Average Values
Section 6-7 Hermitian Operators
6-6 The Postulate for Average Values
Suppose that we had somehow prepared a large number of hydrogen atoms so that they were all in the same, known, stationary state. Then we could measure the distance of the electron from the nucleus once in each atom and average these measurements to obtain an average value. We have already indicated that this average would be given by the sum of all the r values, each multiplied by its frequency of occurrence, which is given by ψ2 dv if ψ is normalized. Since r is a continuous variable, the sum becomes an integral. This is the content of Postulate V When a large number of identical systems have the same state function ψ, the expected average of measurements on the variable M (one measurement per system) is given by
Mav = ψ∗ ˆ
Mψ dτ ψ∗ψ dτ
(6-9)
The denominator is unity if ψ is normalized.
It is important to understand the distinction between average value and eigenvalue as they relate to measurements. A good example is the dipole moment. The dipole moment operator for a system of n charged particles is µ =
n
i=1 zi ri where zi is the charge on the ith particle and ri is its position vector with respect to an arbitrary origin.
(We get this by writing the classical formula and observing that momentum terms do not appear. Hence, the quantum-mechanical operator is the same as the classical expression.) What will the eigenfunctions and eigenvalues of ˆ µ be like?
The charge zi is only a number, while ri is a position operator, which has Dirac delta functions as eigenfunctions. For a hydrogen atom, one eigenfunction of ri would be a delta function at r = 1 a.u., θ = 0, φ = 0. The corresponding eigenvalue for ˆ
µ
would be the dipole moment obtained when a proton and an electron are separated by 1 a.u., clearly a finite number. But “everybody knows” that an unperturbed atom in a stationary state has zero dipole moment. The difficulty is resolved when we recognize that measurement of a variable in postulates IV and V means measuring the value of a variable at a given instant. Hence, we must distinguish between the instantaneous
dipole moment of an atom, which can have any value from among the eigenvalues of ˆµ and the average dipole moment, which is zero for the atom. In everyday scientific discussion, the term “dipole moment” is usually understood to refer to the average
dipole moment. Indeed, the usual measurements of dipole moment are measurements that effectively average over many molecules or long times (in atomic terms) or both.
6-7 Hermitian Operators Let φ and ψ be any square-integrable functions and ˆ A be an operator, all having the same domain. ˆ
A is defined to be hermitian if
ψ∗ ˆ
Aφdv = φ ˆ
A∗ψ∗dv
(6-10)
The integration is over the entire range of each spatial coordinate. Recall that the asterisk signifies reversal of the sign of i in a complex or imaginary term. The hermitian property has important consequences in quantum chemistry.
Chapter 6 Postulates and Theorems of Quantum Mechanics
As an example of a test of an operator by Eq. (6-10), let us take the ψ and φ to be square-integrable functions of x and ˆ A to be i(d/dx). Then the left-hand side of Eq. (6-10) becomes, upon integration by parts,
(6-11)
Since ψ and φ are square integrable, they (and their product) must vanish at infinity, giving the zero term in Eq. (6-11). We now write out the right-hand side of Eq. (6-10):
+∞
+∞
φ(i d/dx)∗ψ∗dx = −i φ(dψ∗/dx) dx
(6-12)
−∞
−∞
where the minus sign comes from carrying out the operation indicated by the asterisk. Equation (6-12) is equal to Eq. (6-11), and so the operator i(d/dx) is hermitian.
Since the effect of i was to introduce a necessary sign reversal, it is apparent that the equality would not result for ˆ A = d/dx. Clearly, any hermitian operator involving a first derivative in any Cartesian coordinate must contain the factor i. The operators for linear momenta (Chapter 2) are examples of this.
It is important to realize that Eq. (6-10) does not imply that ψ∗ ˆ Aφ = φ ˆ A∗ψ∗.
A simple example will make this clearer. Let ˆ
A be the hydrogen atom hamiltonian,
ˆ
√ H = − 1 ∇2 − 1/r, and let φ be the 1s eigenfunction: φ = (1/ π) exp(−r). Also, let
2
√
ψ = 8/π exp(−2r) which is not an eigenfunction of ˆ H . Then, since ˆ H φ = − 1 φ, 2
ψ∗ ˆ H φ = − 1 ψ∗φ
(6-13)
2
But
φH ∗ψ∗ = φ − 1 (1/r2)(d/dr)r2(d/dr) − 1/r 8/π exp(−2r)
(6-14)
2
= φ [(1/r) − 2] 8/π exp(−2r) = [(1/r) − 2] ψ∗φ
(6-15)
(Since ψ has no θ or φ dependence, the parts of ˆ H ∗ that include ∂/∂θ and ∂/∂φ have been omitted in Eq. (6-14).) Here we have two functions, − 1 ψ∗φ and [(1/r) − 2]ψ∗φ.
2
They are obviously different. However, by Eq. (6-10), their integrals are equal since
ˆ
H is hermitian.
6-8 Proof That Eigenvalues of Hermitian OperatorsAre Real
Let ˆ
A be a hermitian operator with a square-integrable eigenfunction ψ. Then
ˆ
Aψ = aψ
(6-16)
Each side of Eq. (6-16) must be expressible as a real and an imaginary part. The real parts must be equal to each other and so must the imaginary parts. Taking the complex
6-6 The Postulate for Average Values
Suppose that we had somehow prepared a large number of hydrogen atoms so that they were all in the same, known, stationary state. Then we could measure the distance of the electron from the nucleus once in each atom and average these measurements to obtain an average value. We have already indicated that this average would be given by the sum of all the r values, each multiplied by its frequency of occurrence, which is given by ψ2 dv if ψ is normalized. Since r is a continuous variable, the sum becomes an integral. This is the content of Postulate V When a large number of identical systems have the same state function ψ, the expected average of measurements on the variable M (one measurement per system) is given by
Mav = ψ∗ ˆ
Mψ dτ ψ∗ψ dτ
(6-9)
The denominator is unity if ψ is normalized.
It is important to understand the distinction between average value and eigenvalue as they relate to measurements. A good example is the dipole moment. The dipole moment operator for a system of n charged particles is µ =
n
i=1 zi ri where zi is the charge on the ith particle and ri is its position vector with respect to an arbitrary origin.
(We get this by writing the classical formula and observing that momentum terms do not appear. Hence, the quantum-mechanical operator is the same as the classical expression.) What will the eigenfunctions and eigenvalues of ˆ µ be like?
The charge zi is only a number, while ri is a position operator, which has Dirac delta functions as eigenfunctions. For a hydrogen atom, one eigenfunction of ri would be a delta function at r = 1 a.u., θ = 0, φ = 0. The corresponding eigenvalue for ˆ
µ
would be the dipole moment obtained when a proton and an electron are separated by 1 a.u., clearly a finite number. But “everybody knows” that an unperturbed atom in a stationary state has zero dipole moment. The difficulty is resolved when we recognize that measurement of a variable in postulates IV and V means measuring the value of a variable at a given instant. Hence, we must distinguish between the instantaneous
dipole moment of an atom, which can have any value from among the eigenvalues of ˆµ and the average dipole moment, which is zero for the atom. In everyday scientific discussion, the term “dipole moment” is usually understood to refer to the average
dipole moment. Indeed, the usual measurements of dipole moment are measurements that effectively average over many molecules or long times (in atomic terms) or both.
6-7 Hermitian Operators Let φ and ψ be any square-integrable functions and ˆ A be an operator, all having the same domain. ˆ
A is defined to be hermitian if
ψ∗ ˆ
Aφdv = φ ˆ
A∗ψ∗dv
(6-10)
The integration is over the entire range of each spatial coordinate. Recall that the asterisk signifies reversal of the sign of i in a complex or imaginary term. The hermitian property has important consequences in quantum chemistry.
Chapter 6 Postulates and Theorems of Quantum Mechanics
As an example of a test of an operator by Eq. (6-10), let us take the ψ and φ to be square-integrable functions of x and ˆ A to be i(d/dx). Then the left-hand side of Eq. (6-10) becomes, upon integration by parts,
(6-11)
Since ψ and φ are square integrable, they (and their product) must vanish at infinity, giving the zero term in Eq. (6-11). We now write out the right-hand side of Eq. (6-10):
+∞
+∞
φ(i d/dx)∗ψ∗dx = −i φ(dψ∗/dx) dx
(6-12)
−∞
−∞
where the minus sign comes from carrying out the operation indicated by the asterisk. Equation (6-12) is equal to Eq. (6-11), and so the operator i(d/dx) is hermitian.
Since the effect of i was to introduce a necessary sign reversal, it is apparent that the equality would not result for ˆ A = d/dx. Clearly, any hermitian operator involving a first derivative in any Cartesian coordinate must contain the factor i. The operators for linear momenta (Chapter 2) are examples of this.
It is important to realize that Eq. (6-10) does not imply that ψ∗ ˆ Aφ = φ ˆ A∗ψ∗.
A simple example will make this clearer. Let ˆ
A be the hydrogen atom hamiltonian,
ˆ
√ H = − 1 ∇2 − 1/r, and let φ be the 1s eigenfunction: φ = (1/ π) exp(−r). Also, let
2
√
ψ = 8/π exp(−2r) which is not an eigenfunction of ˆ H . Then, since ˆ H φ = − 1 φ, 2
ψ∗ ˆ H φ = − 1 ψ∗φ
(6-13)
2
But
φH ∗ψ∗ = φ − 1 (1/r2)(d/dr)r2(d/dr) − 1/r 8/π exp(−2r)
(6-14)
2
= φ [(1/r) − 2] 8/π exp(−2r) = [(1/r) − 2] ψ∗φ
(6-15)
(Since ψ has no θ or φ dependence, the parts of ˆ H ∗ that include ∂/∂θ and ∂/∂φ have been omitted in Eq. (6-14).) Here we have two functions, − 1 ψ∗φ and [(1/r) − 2]ψ∗φ.
2
They are obviously different. However, by Eq. (6-10), their integrals are equal since
ˆ
H is hermitian.
6-8 Proof That Eigenvalues of Hermitian OperatorsAre Real
Let ˆ
A be a hermitian operator with a square-integrable eigenfunction ψ. Then
ˆ
Aψ = aψ
(6-16)
Each side of Eq. (6-16) must be expressible as a real and an imaginary part. The real parts must be equal to each other and so must the imaginary parts. Taking the complex