662
15 The Principles of Quantum Mechanics. I. De Broglie Waves and the Schrödinger Equation
If a system consists of n particles moving in three dimensions, its Hamiltonian
operator is
H −
n
j1
¯
h
2
2m j
∇
2
j + V (q)
(15.2-27)
where ∇ 2
j is the Laplacian operator for the coordinates of particle number j. The
potential energy V is a function of all of the coordinates, which are abbreviated by the
symbol q.
Once the Hamiltonian operator is obtained, the time-dependent Schrödinger equation is abbreviated as in Eq. (15.2-9):
HΨ i ¯
h
∂Ψ
∂t
(15.2-28)
where Ψ depends on all of the coordinates and on t. If we assume the trial solution
Ψ ψ(x 1 , y 1 , z 1 , . . . , x n , y n , z n )η(t)
(15.2-29)
then the time-independent Schrödinger equation can be extracted from the timedependent equation in exactly the same way as in Eqs. (15.2-11) through (15.2-15).
Eigenvalue Equations
The time-independent Schrödinger equation belongs to a class of equations called
eigenvalue equations. The word “eigenvalue” is a partial translation of the German
word Eigenwert. A full translation is “characteristic value.” An eigenvalue equation
has on one side an operator operating on a function, and on the other side a constant
called the eigenvalue multiplying the same function, which is called the eigenfunction.
If
A is a mathematical operator, its eigenvalue equation is
A f n a n f n
(15.2-30)
where f n is the eigenfunction and a n is the eigenvalue. An eigenvalue equation generally has a set of solutions, so we have attached a subscript n to the eigenfunction
and eigenvalue in Eq. (15.2-31) to specify a particular one of the solutions. Solving
an eigenvalue equation means finding not only the set of eigenfunctions that satisfy
the equation, but also the eigenvalue that belongs to each eigenfunction. Two common
cases occur. The first case is that the eigenvalue can take on any value within some
range of values (a continuous spectrum of eigenvalues). The second case is that there
is a discrete set of eigenvalues with the values between the members of the set not
permitted (a discrete spectrum of eigenvalues). The occurrence of a discrete spectrum
of eigenvalues corresponds to quantization.
The time-independent Schrödinger equation is the eigenvalue equation for the
Hamiltonian operator. The coordinate wave function is the eigenfunction of the
Hamiltonian operator, and is often called the energy eigenfunction. The eigenvalue of
the Hamiltonian operator, E, is the value of the energy, and is called the energy eigenvalue. There are other eigenvalue equations that are important in quantum mechanics,
and we will discuss some of them later.
In addition to satisfying the Schrödinger equation a wave function must satisfy other
conditions. Since it represents a wave, we assume that it has the following properties,
15 The Principles of Quantum Mechanics. I. De Broglie Waves and the Schrödinger Equation
If a system consists of n particles moving in three dimensions, its Hamiltonian
operator is
H −
n
j1
¯
h
2
2m j
∇
2
j + V (q)
(15.2-27)
where ∇ 2
j is the Laplacian operator for the coordinates of particle number j. The
potential energy V is a function of all of the coordinates, which are abbreviated by the
symbol q.
Once the Hamiltonian operator is obtained, the time-dependent Schrödinger equation is abbreviated as in Eq. (15.2-9):
HΨ i ¯
h
∂Ψ
∂t
(15.2-28)
where Ψ depends on all of the coordinates and on t. If we assume the trial solution
Ψ ψ(x 1 , y 1 , z 1 , . . . , x n , y n , z n )η(t)
(15.2-29)
then the time-independent Schrödinger equation can be extracted from the timedependent equation in exactly the same way as in Eqs. (15.2-11) through (15.2-15).
Eigenvalue Equations
The time-independent Schrödinger equation belongs to a class of equations called
eigenvalue equations. The word “eigenvalue” is a partial translation of the German
word Eigenwert. A full translation is “characteristic value.” An eigenvalue equation
has on one side an operator operating on a function, and on the other side a constant
called the eigenvalue multiplying the same function, which is called the eigenfunction.
If
A is a mathematical operator, its eigenvalue equation is
A f n a n f n
(15.2-30)
where f n is the eigenfunction and a n is the eigenvalue. An eigenvalue equation generally has a set of solutions, so we have attached a subscript n to the eigenfunction
and eigenvalue in Eq. (15.2-31) to specify a particular one of the solutions. Solving
an eigenvalue equation means finding not only the set of eigenfunctions that satisfy
the equation, but also the eigenvalue that belongs to each eigenfunction. Two common
cases occur. The first case is that the eigenvalue can take on any value within some
range of values (a continuous spectrum of eigenvalues). The second case is that there
is a discrete set of eigenvalues with the values between the members of the set not
permitted (a discrete spectrum of eigenvalues). The occurrence of a discrete spectrum
of eigenvalues corresponds to quantization.
The time-independent Schrödinger equation is the eigenvalue equation for the
Hamiltonian operator. The coordinate wave function is the eigenfunction of the
Hamiltonian operator, and is often called the energy eigenfunction. The eigenvalue of
the Hamiltonian operator, E, is the value of the energy, and is called the energy eigenvalue. There are other eigenvalue equations that are important in quantum mechanics,
and we will discuss some of them later.
In addition to satisfying the Schrödinger equation a wave function must satisfy other
conditions. Since it represents a wave, we assume that it has the following properties,