Section 1-9
Section 1-9 Schr ¨
odinger’s Time-Independent Wave EquationSOLUTION The Heisenberg uncertainty principle gives, for minimum uncertainty E · t = h/4π. E = (6.626 × 10−34 J s)/[(4π)(2 × 10−9 s)]= 2.6 × 10−26 J (2.6 × 10−26J) (5.03 × 1022 cm−1J−1) = 0.001 cm−1 (See Appendix 10 for data.) Larger uncertainty in E shows up as greater line-width in emission spectra.
1-9 Schr ¨odinger’s Time-Independent Wave Equation
Earlier we saw that we needed a wave equation in order to solve for the standing waves pertaining to a particular classical system and its set of boundary conditions. The same need exists for a wave equation to solve for matter waves. Schr¨odinger obtained such an equation by taking the classical time-independent wave equation and substituting de Broglie’s relation for λ. Thus, if ∇2ψ = −(2π/λ)2ψ
(1-47)
and
λ =
h
√
(1-48)
2m(E − V ) then
−(h2/8π2m)∇2 + V (x, y, z) ψ(x, y, z) = Eψ(x, y, z)
(1-49)
Equation (1-49) is Schr¨odinger’s time-independent wave equation for a single particle of mass m moving in the three-dimensional potential field V .
In classical mechanics we have separate equations for wave motion and particle motion, whereas in quantum mechanics, in which the distinction between particles and waves is not clear-cut, we have a single equation—the Schr¨odinger equation. We have seen that the link between the Schr¨odinger equation and the classical wave equation is the de Broglie relation. Let us now compare Schr¨odinger’s equation with the classical equation for particle motion.
Classically, for a particle moving in three dimensions, the total energy is the sum of kinetic and potential energies: (1/2m)(p2 +
+
x
p2y
p2z) + V = E
(1-50)
where px is the momentum in the x coordinate, etc. We have just seen that the analogous Schr¨odinger equation is [writing out Eq. (1-49)]
−h2
∂2 + ∂2 + ∂2 +V (x,y,z) ψ(x,y,z)=Eψ(x,y,z)
(1-51)
8π 2m
∂x2
∂y2
∂z2
It is easily seen that Eq. (1-50) is linked to the quantity in brackets of Eq. (1-51) by a relation associating classical momentum with a partial differential operator: px ←→ (h/2πi)(∂/∂x)
(1-52)
and similarly for py and pz. The relations (1-52) will be seen later to be an important postulate in a formal development of quantum mechanics.
Chapter 1 Classical Waves and the Time-Independent Schr ¨odinger Wave Equation The left-hand side of Eq. (1-50) is called the hamiltonian for the system. For this reason the operator in square brackets on the LHS of Eq. (1-51) is called the hamiltonian
operator6 H . For a given system, we shall see that the construction of H is not difficult.
The difficulty comes in solving Schr¨odinger’s equation, often written as H ψ = Eψ
(1-53)
The classical and quantum-mechanical wave equations that we have discussed are members of a special class of equations called eigenvalue equations. Such equations have the format Op f = cf
(1-54)
where Op is an operator, f is a function, and c is a constant. Thus, eigenvalue equations have the property that operating on a function regenerates the same function times a constant. The function f that satisfies Eq. (1-54) is called an eigenfunction of the operator. The constant c is called the eigenvalue associated with the eigenfunction f . Often, an operator will have a large number of eigenfunctions and eigenvalues of interest associated with it, and so an index is necessary to keep them sorted, viz.
Op fi = cifi
(1-55)
We have already seen an example of this sort of equation, Eq. (1-19) being an eigen function for Eq. (1-18), with eigenvalue −ω2m/T .
The solutions ψ for Schr¨odinger’s equation (1-53), are referred to as eigenfunctions, wavefunctions, or state functions.
EXAMPLE 1-5 a) Show that sin(3.63x) is not an eigenfunction of the operator
d/dx.
b) Show that exp(−3.63ix) is an eigenfunction of the operator d/dx. What is its eigenvalue?
c) Show that 1 sin(3.63x) is an
eigenfunction of the
operator
π
((−h2/8π2m)d2/dx2). What is its eigenvalue?
SOLUTION a) d sin(3.63x) = 3.63 cos(3.63x) = constant times sin(3.63x).
dx
b) d exp(−3.63ix) = −3.63i exp(−3.63ix) = constant times exp(−3.63ix). Eigenvalue =
dx
−3.63i.
c) ((−h2/8π2m)d2/dx2) 1 sin(3.63x) = (−h2/8π2m)(1/π)(3.63) d cos(3.63x)
π
dx
= [(3.63)2h2/8π2m] · (1/π) sin(3.63x) = constant times (1/π) sin(3.63x).
Eigenvalue = (3.63)2h2/8π2m.
6An operator is a symbol telling us to carry out a certain mathematical operation. Thus, d/dx is a differential operator telling us to differentiate anything following it with respect to x. The function 1/x may be viewed as a multiplicative operator. Any function on which it operates gets multiplied by 1/x.
odinger’s Time-Independent Wave EquationSOLUTION The Heisenberg uncertainty principle gives, for minimum uncertainty E · t = h/4π. E = (6.626 × 10−34 J s)/[(4π)(2 × 10−9 s)]= 2.6 × 10−26 J (2.6 × 10−26J) (5.03 × 1022 cm−1J−1) = 0.001 cm−1 (See Appendix 10 for data.) Larger uncertainty in E shows up as greater line-width in emission spectra.
1-9 Schr ¨odinger’s Time-Independent Wave Equation
Earlier we saw that we needed a wave equation in order to solve for the standing waves pertaining to a particular classical system and its set of boundary conditions. The same need exists for a wave equation to solve for matter waves. Schr¨odinger obtained such an equation by taking the classical time-independent wave equation and substituting de Broglie’s relation for λ. Thus, if ∇2ψ = −(2π/λ)2ψ
(1-47)
and
λ =
h
√
(1-48)
2m(E − V ) then
−(h2/8π2m)∇2 + V (x, y, z) ψ(x, y, z) = Eψ(x, y, z)
(1-49)
Equation (1-49) is Schr¨odinger’s time-independent wave equation for a single particle of mass m moving in the three-dimensional potential field V .
In classical mechanics we have separate equations for wave motion and particle motion, whereas in quantum mechanics, in which the distinction between particles and waves is not clear-cut, we have a single equation—the Schr¨odinger equation. We have seen that the link between the Schr¨odinger equation and the classical wave equation is the de Broglie relation. Let us now compare Schr¨odinger’s equation with the classical equation for particle motion.
Classically, for a particle moving in three dimensions, the total energy is the sum of kinetic and potential energies: (1/2m)(p2 +
+
x
p2y
p2z) + V = E
(1-50)
where px is the momentum in the x coordinate, etc. We have just seen that the analogous Schr¨odinger equation is [writing out Eq. (1-49)]
−h2
∂2 + ∂2 + ∂2 +V (x,y,z) ψ(x,y,z)=Eψ(x,y,z)
(1-51)
8π 2m
∂x2
∂y2
∂z2
It is easily seen that Eq. (1-50) is linked to the quantity in brackets of Eq. (1-51) by a relation associating classical momentum with a partial differential operator: px ←→ (h/2πi)(∂/∂x)
(1-52)
and similarly for py and pz. The relations (1-52) will be seen later to be an important postulate in a formal development of quantum mechanics.
Chapter 1 Classical Waves and the Time-Independent Schr ¨odinger Wave Equation The left-hand side of Eq. (1-50) is called the hamiltonian for the system. For this reason the operator in square brackets on the LHS of Eq. (1-51) is called the hamiltonian
operator6 H . For a given system, we shall see that the construction of H is not difficult.
The difficulty comes in solving Schr¨odinger’s equation, often written as H ψ = Eψ
(1-53)
The classical and quantum-mechanical wave equations that we have discussed are members of a special class of equations called eigenvalue equations. Such equations have the format Op f = cf
(1-54)
where Op is an operator, f is a function, and c is a constant. Thus, eigenvalue equations have the property that operating on a function regenerates the same function times a constant. The function f that satisfies Eq. (1-54) is called an eigenfunction of the operator. The constant c is called the eigenvalue associated with the eigenfunction f . Often, an operator will have a large number of eigenfunctions and eigenvalues of interest associated with it, and so an index is necessary to keep them sorted, viz.
Op fi = cifi
(1-55)
We have already seen an example of this sort of equation, Eq. (1-19) being an eigen function for Eq. (1-18), with eigenvalue −ω2m/T .
The solutions ψ for Schr¨odinger’s equation (1-53), are referred to as eigenfunctions, wavefunctions, or state functions.
EXAMPLE 1-5 a) Show that sin(3.63x) is not an eigenfunction of the operator
d/dx.
b) Show that exp(−3.63ix) is an eigenfunction of the operator d/dx. What is its eigenvalue?
c) Show that 1 sin(3.63x) is an
eigenfunction of the
operator
π
((−h2/8π2m)d2/dx2). What is its eigenvalue?
SOLUTION a) d sin(3.63x) = 3.63 cos(3.63x) = constant times sin(3.63x).
dx
b) d exp(−3.63ix) = −3.63i exp(−3.63ix) = constant times exp(−3.63ix). Eigenvalue =
dx
−3.63i.
c) ((−h2/8π2m)d2/dx2) 1 sin(3.63x) = (−h2/8π2m)(1/π)(3.63) d cos(3.63x)
π
dx
= [(3.63)2h2/8π2m] · (1/π) sin(3.63x) = constant times (1/π) sin(3.63x).
Eigenvalue = (3.63)2h2/8π2m.
6An operator is a symbol telling us to carry out a certain mathematical operation. Thus, d/dx is a differential operator telling us to differentiate anything following it with respect to x. The function 1/x may be viewed as a multiplicative operator. Any function on which it operates gets multiplied by 1/x.