determined by combining thePlanck–Einstein relation(E = ¯hω) with the momentum-energy relation
p= E c = ¯hω
c . (3.3)
Since the frequency (f) of a wave is related to its wavelength (λ) and speed (c) by the equationf = λc, the momentum may be written as
p= ¯hω
c = ¯h(2πf) c = ¯h
2πλc c
= ¯h2π λ .
The definition of wavenumber (k≡ 2πλ) makes this
p= ¯hk. (3.4)
This equation is known as de Broglie’s relation, and it represents the mixing of wave and particle behavior into the concept of “wave-particle duality.”
Since momentum is the product of mass and velocity (p = mv) in the nonrelativistic case, the classical equation for kinetic energy is
KE=1
2mv2= p2
2m (3.5)
and substitutinghk¯ for momentum (Eq. 3.4) gives KE= ¯h2k2
2m . (3.6)
Now write the total energy (E) as the sum of the kinetic energy (KE) plus the potential energy (V):
E=KE+V= ¯h2k2
2m +V, (3.7)
and sinceE= ¯hω(Eq. 3.1), the total energy is given by E= ¯hω= ¯h2k2
2m +V. (3.8)
This equation provides the foundation for the Schr¨odinger equation when applied to a quantum wavefunction(x,t).
To get fromEq. 3.8to the Schr¨odinger equation, one path is to assume that the quantum wavefunction has the form of a wave for which the surfaces of
constant phase are flat planes.2For aplane wavepropagating in the positive x-direction, the wavefunction is given by
(x,t)=Aei(kx−ωt), (3.9)
in whichArepresents the wave’s amplitude,kis the wavenumber, andωis the angular frequency of the wave.
With this expression for , taking temporal and spatial derivatives is straightforward (and helpful in getting from Eq. 3.8 to the Schr¨odinger equation). Starting with the first partial derivative of(x,t)with respect to time (t),
∂(x,t)
∂t =∂$
Aei(kx−ωt)%
∂t = −iω&
Aei(kx−ωt)
'= −iω(x,t). (3.10)
So for the plane-wave function ofEq. 3.9, taking the first partial derivative with respect to time has the effect of returning the original wavefunction multiplied by−iω:
∂
∂t = −iω, (3.11)
which means that you can writeωas ω= 1
−i
∂
∂t =i1
∂
∂t , (3.12)
in which the relation 1i = −(i)(i)i = −ihas been used.
Now consider what happens when you take the first partial derivative of (x,t)with respect to space (xin this case):
∂(x,t)
∂x =∂$
Aei(kx−ωt)%
∂x =ik
&
Aei(kx−ωt)
'=ik(x,t). (3.13)
So taking the first partial derivative of the plane-wave function with respect to distance (x) has the effect of returning the original wavefunction multiplied byik:
∂
∂x =ik. (3.14)
It’s also helpful to note the effect of taking the second partial spatial derivative of the plane-wave function, which gives
∂2(x,t)
∂x2 = ∂$
ikAei(kx−ωt)%
∂x =ik
&
ikAei(kx−ωt)
'= −k2(x,t), (3.15)
2If you’re unfamiliar with plane waves, you can see a sketch of the planes of constant phase in Fig. 3.4inSection 3.4.
which means that taking the second partial derivative with respect toxhas the effect of returning the original wavefunction multiplied by−k2:
∂2
∂x2 = −k2. (3.16)
So just as the angular frequencyωmay be written in terms of the wavefunction and its temporal partial derivative ∂∂t (Eq. 3.12), the square of the wavenumber k may be written in term of and its second spatial partial derivative ∂∂x22 :
k2= −1
∂2
∂x2. (3.17)
What good has it done to writeωandk2in terms of the wavefunction and its derivatives? To understand that, look back atEq. 3.8, and note that it includes a factor ofωon the left side and a factor ofk2on the right side of the second equals sign. Substituting the expression forωfromEq. 3.12 into the left side gives
E= ¯hω= ¯h
i1
∂
∂t
=ih¯1
∂
∂t . (3.18)
Likewise, substituting the expression fork2fromEq. 3.17into the right side ofEq. 3.8gives
¯ h2k2
2m +V= ¯h2 2m
"
−1
∂2
∂x2
#
+V, (3.19)
which makes the equation for total energy look like this:
ih¯1
∂[(x,t)]
∂t = − ¯h2 2m
1
∂2[(x,t)]
∂x2 +V, (3.20)
and multiplying through by the wavefunction(x,t)yields ih¯∂[(x,t)]
∂t = − ¯h2 2m
∂2[(x,t)]
∂x2 +V[(x,t)]. (3.21) This is the most common form of the one-dimensional time-dependent Schr¨odinger equation. The physical meaning of this equation and each of its terms is discussed in this chapter, but before getting to that, you should consider how we got here. Writing the total energy as the sum of the kinetic energy and the potential energy is perfectly general, but to get toEq. 3.21, we used the expression for a plane wave. Specifically,Eq. 3.12forωandEq. 3.17 for k2 resulted from the temporal and spatial derivatives of the plane-wave
function (Eq. 3.9). Why should we expect this equation to hold for quantum wavefunctions of other forms?
One answer is this: it works. That is, wavefunctions that are solutions to the Schr¨odinger equation lead to predictions that agree with laboratory measure- ments of quantum observables such as position, momentum, and energy.
If it seems surprising that an equation based on a simple plane-wave function describes the behavior of particles and systems that have little in common with plane waves, note that the Schr¨odinger equation is linear, which means that the terms involving the wavefunction, such as ∂[(x,t)]∂t , ∂2[(x,t)]∂x2 , andV(x,t), are all raised to the first power.3 As you may recall, a linear equation has the supremely useful characteristic that superposition works, which guarantees that combinations of solutions are also solutions. And since plane waves are solutions to the Schr¨odinger equation, the linear nature of the equation means that superpositions of plane waves are also solutions.
By judicious combination of plane waves, a variety of quantum wavefunctions may be synthesized, just as a variety of functions may be synthesized from the sine and cosine functions in Fourier analysis.
To understand why that works, consider the wavefunction of a quantum particle that is localized over some region of the x-axis. Since a single- frequency plane wave extends to infinity in both directions (±x), it’s clear that additional frequency components are needed to restrict the particle’s wavefunction to the desired region. Combining those components in just the right proportion allows you to form a “wave packet” with amplitude that rolls off with distance from the center of the packet.
To form a wavefunction from a finite number (N) of discrete plane-wave components, a weighted linear combination may be used:
(x,t)=A1ei(k1x−ω1t)+A2ei(k2x−ω2t)+ · · · +ANei(kNx−ωNt)
=nN=1Anei(knx−ωnt), (3.22) in which An,kn, and ωn represent the amplitude, wave number, and angu- lar frequency of the nth plane-wave component, respectively. Note that the constants An determine the “amount” of each plane wave included in the mix.
3Remember that the second-order derivative∂2
∂x2 represents the change in the slope ofwith respect tox, which is not the same as the square of the slope(∂∂x)2. So∂2
∂x2 is a second-order derivative, but it’s raised to the first power in the Schr¨odinger equation.
Alternatively, a wavefunction satisfying the Schr¨odinger equation can be synthesized using a continuous spectrum of plane waves:
(x,t)= ∞
−∞A(k)ei(kx−ωt)dk, (3.23) in which the summation of Eq. 3.22 is now an integral and the discrete amplitudesAn have been replaced by a continuous function of wavenumber A(k). As in the discrete case, this function is related to the amplitude of the plane-wave components as a function of wavenumber. Specifically, in the continuous caseA(k)represents the amplitude per unit wavenumber.
And just as in the case of an individual plane wave, taking the first- order time derivative and second-order spatial derivative of wavefunctions synthesized from combinations of plane waves leads to the Schr¨odinger equation.
A very common and useful version ofEq. 3.23can be obtained by pulling a constant factor of 1/√
2πout of the weighting functionA(k)and setting the time to an initial reference time (t=0):
ψ(x)=(x, 0)= 1
√2π ∞
−∞φ (k)eikxdk. (3.24) This version makes clear the Fourier-transform relationship between the position-based wavefunction ψ(x)and the wavenumber-based wavefunction φ (k), which plays an important role inChapters 4and5. You can read about Fourier transforms inSection 4.4.
Before considering exactly what the Schr¨odinger equation tells you about the behavior of quantum wavefunctions, it’s worthwhile to consider another form of the Schr¨odinger equation that you’re very likely to encounter in textbooks on quantum mechanics. That version ofEq. 3.21looks like this:
ih¯∂
∂t =H. (3.25)
In this equation, H represents the “Hamiltonian,” or total-energy operator.
Equating the right sides of this equation andEq. 3.21gives H = − ¯h2
2m
∂2
∂x2 +V,
which means that theHamiltonian operatoris equivalent to H≡ − ¯h2
2m
∂2
∂x2+V. (3.26)
To see why this makes sense, use the relations p = ¯hk andE = ¯hωto rewrite the plane-wave function in terms of momentum (p) and energy (E)
(x,t)=Aei(kx−ωt)=Aei
p
¯ hx−E¯ht
=Ae¯hi(px−Et), (3.27)
and then take the first-order spatial derivative:
∂
∂x = i
¯ hp
Aehi¯(px−Et)= i
¯ hp
or
p = ¯h i
∂
∂x = −ih¯∂
∂x. (3.28)
This suggests that the (one-dimensional) differential operator associated with momentum may be written as
p= −ih¯ ∂
∂x. (3.29)
This is a very useful relation in its own right, but for now you can use it to justify Eq. 3.26for the Hamiltonian operator. To do that, write an operator version of the classical total-energy equationE= 2mp2 +V:
H= (p)2 2m +V=
−ih¯∂x∂ 2
2m +V, (3.30)
in whichHis an operator associated with the total energyE.
Now recall that, unlike the square of an algebraic quantity, the square of an operator is formed by applying the operator twice. For example, the square of operatorOoperating on functionis
( O)2 =O( O), so
(p)2 =p(p)= −ih¯ ∂
∂x
−ih¯∂
∂x
=i2h¯2∂2
∂x2 = −¯h2∂2
∂x2 . (3.31)
Thus the(p)2operator may be written as (p)2= −¯h2 ∂2
∂x2 (3.32)
and plugging this expression intoEq. 3.30gives H=(p)2
2m +V = −h¯2∂2
∂x2
2m +V
=−¯h2 2m
∂2
∂x2 +V, in agreement withEq. 3.26.
In the next section, you can read more about the meaning of each term in the Schr¨odinger equation as well as the meaning of the equation as a whole. And if you’d like to see some alternative approaches to “deriving” the Schr¨odinger equation, on the book’s website you can find descriptions of the “probability flow” approach and the “path integral” approach along with links to helpful websites for those approaches.