15.2 The Schrödinger Equation
659
Exercise 15.2
Carry out the algebra to obtain Eq. (15.2-6) from Eq. (15.2-2).
The left-hand side of Eq. (15.2-6) is commonly abbreviated by defining
H −
¯
h
2
2m
d 2
dx 2 + V (x)
(15.2-7)
so that we abbreviate the Schrödinger equation in the form:
Hψ Eψ
(15.2-8)
The symbol
H is a mathematical operator, because it stands for the carrying out of
mathematical operations on the function ψ. It is called the Hamiltonian operator.
The Time-Dependent Schrödinger Equation
The time-dependent Schrödinger equation is taken as one of the postulates (fundamental
assumptions) of quantum theory:
HΨ i ¯
h
∂Ψ
∂t
(15.2-9)
where i is the imaginary unit, defined to equal the square root of −1:
i
√ −1
(15.2-10)
and where the
H operator is the same as in the time-independent equation. The function
Ψ is the time-dependent wave function and represents the displacement of a de Broglie
wave as a function of position and time. In this chapter and the next we will use a capital
psi (Ψ) for a time-dependent wave function and a lower-case psi (ψ) for a coordinate
factor (coordinate wave function).
There is no way to obtain the time-dependent Schrödinger equation from a classical wave equation. The classical wave equation of a vibrating string, Eq. (14.3-3), is
second order in time. It requires two initial conditions (an initial position and an initial
velocity) to make a general solution apply to a specific case. The uncertainty principle of quantum mechanics (to be discussed later) implies that positions and velocities
cannot be specified simultaneously with arbitrary accuracy. For this reason only one
initial condition is possible, which requires the Schrödinger equation to be first order
in time. The fact that the equation is first order in time also requires that the imaginary
unit i must occur in the equation in order for oscillatory solutions to exist.
The time-independent Schrödinger equation can be obtained from the time-dependent
equation by separation of variables. For motion in the x direction, we assume the trial
function
Ψ(x, t) ψ(x)η(t)
(15.2-11)
659
Exercise 15.2
Carry out the algebra to obtain Eq. (15.2-6) from Eq. (15.2-2).
The left-hand side of Eq. (15.2-6) is commonly abbreviated by defining
H −
¯
h
2
2m
d 2
dx 2 + V (x)
(15.2-7)
so that we abbreviate the Schrödinger equation in the form:
Hψ Eψ
(15.2-8)
The symbol
H is a mathematical operator, because it stands for the carrying out of
mathematical operations on the function ψ. It is called the Hamiltonian operator.
The Time-Dependent Schrödinger Equation
The time-dependent Schrödinger equation is taken as one of the postulates (fundamental
assumptions) of quantum theory:
HΨ i ¯
h
∂Ψ
∂t
(15.2-9)
where i is the imaginary unit, defined to equal the square root of −1:
i
√ −1
(15.2-10)
and where the
H operator is the same as in the time-independent equation. The function
Ψ is the time-dependent wave function and represents the displacement of a de Broglie
wave as a function of position and time. In this chapter and the next we will use a capital
psi (Ψ) for a time-dependent wave function and a lower-case psi (ψ) for a coordinate
factor (coordinate wave function).
There is no way to obtain the time-dependent Schrödinger equation from a classical wave equation. The classical wave equation of a vibrating string, Eq. (14.3-3), is
second order in time. It requires two initial conditions (an initial position and an initial
velocity) to make a general solution apply to a specific case. The uncertainty principle of quantum mechanics (to be discussed later) implies that positions and velocities
cannot be specified simultaneously with arbitrary accuracy. For this reason only one
initial condition is possible, which requires the Schrödinger equation to be first order
in time. The fact that the equation is first order in time also requires that the imaginary
unit i must occur in the equation in order for oscillatory solutions to exist.
The time-independent Schrödinger equation can be obtained from the time-dependent
equation by separation of variables. For motion in the x direction, we assume the trial
function
Ψ(x, t) ψ(x)η(t)
(15.2-11)