What the Schr¨odinger Equation Means

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.

∂

∂ t

: The quantum wavefunction(x,t)is a function of both time and space, so this term represents the change in the wavefunction over time only (which is why it’s a partial derivative). In a graph of the wavefunction at a given location as a function of time, this term is the slope of the graph. To determine the dimensions of this term, note that the one- dimensional quantum wavefunction represents a probability density amplitude (which you can read about inChapter 4), the square of which has dimensions of probability per unit length. This is equivalent to_m¹ in the SI system, since probability is dimensionless. And if²has units of _m¹, then must have units of√¹

m, which means that^∂_∂t has SI units of _s√¹

m.

i

: The numerical value of the imaginary unitiis√

−1, as described in Section 1.4. As an operator, multiplication byihas the effect of causing a 90^◦rotation in the complex plane (Fig. 1.7), moving numbers from the positive real axis to the positive imaginary axis, or from the positive imaginary axis to the negative real axis, for example. The presence of i in the Schr¨odinger equation means that the quantum wavefunction solutions may be complex, and this significantly impacts the result of combining wavefunctions, as you can see in Chapters 4 and5. The factoriis dimensionless.

¯

h

: The modified Planck constanth¯is the Planck constanthdivided by 2π.

Just ashis the constant of proportionality between the energy (E) and frequency (f) of a photon (E=hf),h¯is the constant of proportionality between total energy (E) and angular frequency (ω), and between momentum (p) and wavenumber (k) in quantum wavefunctions, as shown in the equationsE= ¯hωandp= ¯hk.

These two equations account for the presence of the modified Planck constant in the Schr¨odinger equation. The modified Planck constanth¯ appears in the numerator of the factor multiplying^∂_∂t on one side of the Schr¨odinger equation because it appears in the total-energy equation E = ¯hω, and the square ofh¯ appears in the numerator of the factor multiplying ^∂_∂x²₂ because it appears in the momentum equationp= ¯hk, which gives rise to the kinetic-energy expressionKE=^(hk)¯_2m².

The Planck constanthhas dimensions of energy per unit frequency, so its SI units are Joules per Hertz (equivalent to Js or m²kg/s), while

¯

hhas dimensions of Joules per Hertz per radian (equivalent to Js/rad or m²kg/s rad). The numerical values of these constants in the SI system areh=6.62607×10⁻³⁴Js andh¯ =1.05457×10⁻³⁴Js/rad.

m

: The mass of the particle or system associated with the quantum wavefunction (x,t) is a measure of inertia, that is, resistance to acceleration. In the SI system, mass has units of kilograms.

∂

²

∂x

2 : This second-derivative term represents the curvature of the wavefunc- tion over space (that is, over x in the one-dimensional case). Since (x,t)is a function of both space and time, the first partial derivative^∂_∂x gives the change of the wavefunction over space (the slope of the wavefunction plotted againstx), and the second partial derivative ^∂_∂x²2

gives the change in the slope of the wavefunction over space (that is, the curvature of the wavefunction).

Since(x,t)has SI units of √¹

m, as described earlier, the term ^∂_∂x²₂ has units of ¹

m²√

m= _(m)¹5/2.

V

: The potential energy of the system may vary over space and time, in which case you’ll see this term written as V(x,t) for the one- dimensional case or asV(r,t)in the three-dimensional case. Note that some physics texts useVto denote the electrostatic potential (potential energy per unit charge, with units of Joules per Coulomb or volts), but in quantum mechanics texts the words “potential” and “potential energy”

tend to be used interchangeably.

Unlike classical mechanics, in which the potential, kinetic, and total energy have precise values, and in which the potential energy cannot exceed the total energy, in quantum mechanics only the average or expectation value of the energy may be determined, and a particle’s total energy may be less than the potential energy in some regions.

The behavior of quantum wavefunctions in these classically “unal- lowed” regions (in whichE<V) is very different from their behavior in classically “allowed” regions (in whichE ≥ V). As you’ll see in the next section of this chapter, for the “stationary solutions” of the time-independent version of the Schr¨odinger equation, the difference between the total energy and the potential energy determines the wave- length for oscillating solutions in classically allowed regions and the rate of decay for evanescent solutions in classically unallowed regions.

As you may have guessed, the potential-energy term in the Schr¨odinger equation has dimensions of energy and SI units of Joules (equivalent to kg m²/s²).

So the individual terms of the Schr¨odinger equation are readily understand- able, but the real power of this equation comes from the relationship between

those terms. Taken together, the terms of the Schr¨odinger equation form a parabolic second-order partial differential equation. Here’s why each of those terms applies:

Differential because the equation involves thechangein the wavefunction (that is, the derivatives of(x,t)over space and time);

Partial because the wavefunction(x,t)depends on both space (x) and time (t);

Second-order because the highest derivative (^∂_∂x²₂) in the equation is a second derivative;

Parabolic because the combination of a first-order differential term (^∂_∂t) and a second-order differential term (^∂_∂x²₂) is analogous to the combination of a first-order algebraic term (y) and a second- order algebraic term (x²) in the equation of a parabola (y=cx²).

These terms describe what the Schr¨odinger equation is, but what does it mean? To understand that, you may find it helpful to consider a well-known equation in classical physics:

∂[f(x,t)]

∂t =D∂²[f(x,t)]

∂x² . (3.33)

This one-dimensional “diffusion” equation⁴describes the behavior of a quan- tity f(x,t) with spatial distribution that may evolve over time, such as the concentration of a substance or the temperature of a fluid. In the diffusion equation, the proportionality factor “D” between the first-order time derivative and the second-order space derivative represents the diffusion coefficient.

To see the similarity between the classical diffusion equation and the Schr¨odinger equation, consider the case in which the potential energy (V) is zero, and writeEq. 3.21as

∂[(x,t)]

∂t = ih¯ 2m

∂²[(x,t)]

∂x² . (3.34)

Comparing this form of the Schr¨odinger equation to the diffusion equation, you can see that both relate the first-order time derivative of a function to the second-order spatial derivative of that function. But as you might expect, the presence of the “i” factor in the Schr¨odinger equation has important implications for the wavefunctions that are solutions of that equation, and you can read about those implications in Chapters 4 and 5. But for now, you should make sure you understand the fundamental relationship in both

4This equation is also called the heat equation or Fick’s second law.

Positive Curvature (slope is positive and getting steeper)

Negative Curvature (slope is positive and getting less steep)

Negative Curvature (slope is negative and getting steeper)

Positive Curvature (slope is negative and getting less steep) Inflection

Points

f(x,t) at time

t = 0

x

Figure 3.2 Regions of positive and negative curvature for peaked waveform.

of these equations: the evolution of the waveform over time is proportional to the curvature of the waveform over space.

And why should the rate of change of a function be related to the spatial curvature of that function? To understand that, consider the functionf(x,t) shown inFig. 3.2for timet=0. This function could represent, for example, the initial temperature distribution of a fluid with a warm spot in the region of x=0. To determine how this temperature distribution will evolve over time, the diffusion equation tells you to consider the curvature of the wavefunction in various regions.

As you can see in the figure, this function has a maximum atx = 0 and inflection points⁵ atx = −3 andx = +3. For the region to the left of the inflection point atx= −3, the slope of the function (^∂f_∂x) is positive and getting more positive asxincreases, which means the curvature in this region (that is, thechangein the slope ^∂_∂x^2f₂) is positive. Likewise, to the right of the inflection point atx= +3, the slope of the function is negative but getting less negative with increasingx, meaning that the curvature (again, the change in the slope) is positive in this region as well.

Now consider the regions betweenx= −3 andx=0 and betweenx=0 andx= +3. Betweenx= −3 andx=0, the slope of the function is positive

5An inflection point is a location at which the sign of the curvature changes.

Amplitude increases over time in regions of positive curvature

Amplitude decreases over time in regions of negative curvature

Waveform at later time

Waveform at time t = 0 f(x,t)

x

Figure 3.3 Time evolution for regions of positive and negative curvature.

but becoming less steep with increasingx, so the curvature in this region is negative. And betweenx=0 andx= +3, the slope is negative and becoming steeper with increasingx, so the curvature in this region is also negative.

And here’s the payoff: since the diffusion equation tells you that the time rate of change of the functionf(x,t) is proportional to the curvature of the function, the function will evolve as shown inFig. 3.3.

As you can see in that figure, the functionf(x,t)will increase in regions of positive curvature (x < −3 and x > +3) and will decrease in regions of negative curvature (−3 < x < +3). If f(x,t)represents temperature, for example, this is exactly what you’d expect as the energy from the initially warm region diffuses into the cooler neighboring regions.

So given the similarity between the Schr¨odinger equation and the classical diffusion equation, does that mean that all quantum particles and systems will somehow “diffuse” or spread out in space as time passes? If so, exactly what is it that’s spreading out?

The answer to the first of these questions is “Sometimes, but not always.”

The reason for that answer can be understood by considering an important difference between the Schr¨odinger equation and the diffusion equation. That difference is the factor of “i” in the Schr¨odinger equation, which means that the wavefunction () can be complex. And as you’ll see inChapters 4and 5, complex wavefunctions may exhibit wavelike (oscillatory) behavior rather than diffusing under some circumstances.

As for the question about what’s spreading out (or oscillating), for that answer we turn to Max Born, whose 1926 interpretation of the wavefunction as a probability amplitude is now widely accepted and is a fundamental precept of the Copenhagen interpretation of quantum mechanics, which you can read about inChapter 4. According to the “Born rule,” the modulus squared (||²=^∗) of a particle’s position-space wavefunction gives the particle’s position probability density function (that is, the probability per unit length in the one-dimensional case). This means that the integral of the position probability density function over any spatial interval gives the probability of finding the particle within that interval. So when the wavefunction oscillates or diffuses, it’s the probability distribution that’s changing.

Here’s another propitious characteristic of the Schr¨odinger equation: the time derivative ^∂_∂t is first-order, which differs from the second-order time derivative of the classical wave equation. Why is that helpful? Because a first- order time derivative tells you how fast the wavefunction itself is changing over time, which means that knowledge of the wavefunction at some instant in time completely specifies the state of the particle or system at all future times. That’s consistent with the principle that the wavefunctions that satisfy the Schr¨odinger equation represent “all you can ever know” about the state of a particle or system.

But if you’re a fan of the classical wave equation with its second-order time and spatial derivatives, you may be wondering whether it’s useful to take another time derivative of the Schr¨odinger equation. That’s certainly possible, but recall that taking another time derivative would pull down another factor ofωfrom the plane-wave functione^i(kx−ωt), andωis proportional toEby de Broglie’s relation (E = ¯hω). That means that the resulting equation would include the particle’s energy as a coefficient of the time-derivative term.

You may be thinking, “But don’t all equations of motion depend on energy?” Definitely not, as you can see by considering Newton’s Second Law:

F = ma, better written as a = F/m. This says that the acceleration of an object is directly proportional to the vector sum of the forces acting on it, and inversely proportional to the object’s mass. But in classical physics the acceler- ation does not depend on the energy, momentum, or velocity of the object. So if the Schr¨odinger equation is to serve a purpose in quantum mechanics similar to that of Newton’s Second Law in classical mechanics, the time evolution of the wavefunction shouldn’t depend on the particle or system’s energy or momentum. Hence the time derivative cannot be of second order.

So although the Schr¨odinger equation can’t be derived from first principles, the form of the equation does make sense. More importantly, it gives results

that predict and describe the behavior of quantum particles and systems over space and time. But one very useful form of the Schr¨odinger equation is independent of time, and that version is the subject of the next section.