Chapter 8

Differential Equations

To paraphrase the Greek philosopher Heraclitus,“The only thing constant is change.” Differential equations describe mathematically how things in the physical world change—both in time and in space. While the solution of an algebraic equation is usually a number, the solution of a differential equation gives a function.

In this chapter, we consider ordinary differential equations (ODEs), which involve just two variables, say, an independent variable image and a dependent variable image. Later we will take up partial differential equations, which have two or more independent variables. The most general ordinary differential equation is a relationship of the form

image (8.1)

in which the object is to solve for image, consistent with one or more specified boundary conditions. The order of a differential equation is determined by the highest-order derivative. For a large number of applications in physics, chemistry, and engineering, it suffices to consider first-and second-order differential equations, with no higher derivatives than image. In this chapter, we will deal with linear ODEs of the general form

image (8.2)

where image, and image are known functions. When image, the equation is said to be homogeneous, otherwise inhomogeneous.

8.1 First-Order Differential Equations

A first-order equation of the form

image (8.3)

can be solved by separation of variables. We rearrange the equation to

image (8.4)

and then integrate to give

image (8.5)

After exponentiating the equation (making it the exponent of image), we obtain

image (8.6)

The constant is determined by an initial or boundary condition.

For the inhomogeneous analog of (8.3):

image (8.7)

separation of variables can be done after introducing an integrating factor. Note first that

image (8.8)

Thus Eq. (8.7) can be arranged to

image (8.9)

The solution is

image (8.10)

This will look a lot less intimidating once the function image is explicitly specified.

We will consider next a few illustrative examples of first-order differential equations of physical significance. In several cases, independent variable image will be the time image.

A radioactive element disintegrates at a rate which is independent of any external conditions. The number of disintegrations occurring per unit time is proportional to the number of atoms originally present. This can be formulated as a first-order differential equation in time:

image (8.11)

The rate constantimage is a characteristic of the radioactive element. The minus sign reflects the fact that the number of atoms imagedecreases with time. Equation (8.11) is easily solved by separation of variables. We find

image (8.12)

We now evaluate the indefinite integral of each side of the equation:

image (8.13)

The constant of integration is, in this case, determined by the initial condition: at time image, there are image atoms. Exponentiating, we obtain

image (8.14)

which is the famous equation for exponential decay. Radioactive elements are usually characterized by their half-lifeimage, the time it takes for half the atoms to disintegrate. Setting image, we find the relation between half-life and decay constant:

image (8.15)

The result for radioactive decay is easily adapted to describe organic growth, the rate at which the population of a bacterial culture will increase given unlimited nutrition.We simply change image to image in Eq. (8.11), which leads to exponential growth:

image (8.16)

This remains valid until the source of nutrition runs out or the population density becomes intolerable.

A bimolecular chemical reaction such as image often follows a rate law

image (8.17)

in which the increase in concentration of the reaction product C is proportional to the product (in the sense of image) of concentrations of the reactants A and B. To put this into more concrete form, assume that the initial concentrations at image are image, image. At a later time image, the concentration [C] grows to image while [A] and [B] are reduced to image. Equation (8.17) then reduces to

image (8.18)

Separation of variables followed by integration gives

image (8.19)

Since image when image, the constant equals image and the solution can be written

image (8.20)

More challenging is the case when image, different from image. If you are adventurous, you can work out the solution

image (8.21)

Problem 8.1.1

Find the solution to the differential equation

image

satisfying the boundary condition image.

Problem 8.1.2

Solve the differential equation

image

You will need to introduce an integrating factor.

Bernoulli’s differential equation:

image (8.22)

is a well-known nonlinear equation that can be reduced to a linear equation. Multiplying by image, we obtain

image (8.23)

This reduces to a linear equation in the variable image:

image (8.24)

Using the integrating factor image, the general solution, after reverting to the original variable image, is given by

image (8.25)

The equation is already linear if image and separable, as well, if image.

8.2 Numerical Solutions

Numerical methods can be used to obtain approximate solutions for differential equations that cannot be treated analytically. The most rudimentary procedure for first-order equations is the Euler method. Not, in itself, very accurate it serves as a starting point for more advanced techniques. Suppose we seek a solution of a first-order equation of the form image, subject to the boundary condition image. The first approximation for the value of image at the point image might be suggested by the starting point in the definition of a derivative

image (8.26)

Choose a value for the size of each step between successive image-values, image, so that image. Thus image. Each successive step in the Euler method is then determined by the recursive relations:

image (8.27)

This procedure is illustrated by the chain of red1 segments in Figure 8.1, with the blue curve representing the hypothetical exact solution. Clearly, the accuracy degrades with increasing image, although it can be improved by decreasing the interval size image.

image

Figure 8.1 Pictorial representation of the Euler method.

Runge-Kutta methods provide more advanced approaches to numerical integration of ordinary differential equations. The most common variant, known as RK4, is an elaboration of the Euler method using four-term interpolations to determine successive points on the solution curve. Intermediate values of the functions image between image and image enter into the computation of successive coefficients. The RK4 method is based on the following equations:

image (8.28)

with

image (8.29)

image (8.30)

image (8.31)

image (8.32)

When image is a function of image alone, the differential equation is essentially just an integration and the procedure reduces to Simpson’s rule. The RK4 method has an accumulated error of order image, whereas the Euler method had an error which was first order in image.

Mathematica is particularly adept at numerical solutions of differential equations, automatically choosing the most appropriate algorithm. For example, a first-order equation of the type discussed above can be programed using a command something like: NDSolve[{y[x]==, y[a]==…},{x, a, b}] .

8.3 AC Circuits

A number of instructive applications of differential equations concern alternating-current (AC) circuits containing resistance, inductance, capacitance and an oscillating voltage source, as represented in Figure 8.2. In the simplest case, consider a circuit with resistance image and voltage (or emf) image. The current image is determined by Ohm’s law:

image (8.33)

The standard units are amperes for image, volts for image, and ohms for image. The other relevant units are henrys for image and farads for image. Ohm’s law is true even for an AC circuit, in which the voltage varies sinusoidally with time, say

image (8.34)

The current is then given by

image (8.35)

Thus the current through a resistance oscillates in phase with the voltage. The frequency image is expressed in units of rad/s. It is more common to measure frequency image in cycles/s, a unit called the hertz (Hz). Since one cycle traces out image radians, the two measures of frequency are related by

image (8.36)

Thus your 60 Hz household voltage has an angular frequency of image.

image

Figure 8.2 Series RLC circuit powered by an oscillating emf.

The voltage change across an inductance is given by image. Thus for a circuit with inductance, but negligible resistance, the analog of Ohm’s law is

image (8.37)

With an oscillating voltage (8.34), with an initial current image, this equation is easily integrated to give

image (8.38)

where the inductive reactanceimage has the same units as image. For a DC voltage image, an inductor behaves just like an ordinary conductor. Note that the current in Eq. (8.38) is image out of phase with the voltage image. Specifically, for a pure inductance, the current lags the voltage by image. Alternatively stated, the voltage leads the current by image. Physically, this reflects the fact that the inductor builds up an opposing emf (by Lenz’s law) in response to an increase in current.

For a circuit with capacitance image, the relevant relation is

image (8.39)

where image (in coulombs) is the charge on the capacitor. In a DC circuit, no current passes through a capacitor. For an AC circuit, however, with the current being given by image, we find

image (8.40)

For an AC voltage (8.34), the equation integrates to give

image (8.41)

where the capacitive reactance is defined by image. This shows that for a pure capacitance, the current leads the voltage by image. The mnemonic “ELI the ICEman” summarizes the phase relationships for inductance and capacitance: for image, image leads image while for image, image leads image.

In an electrical circuit with both resistance and inductance, the current and voltage are related by

image (8.42)

Suppose at time image, while the current has the value image, the voltage image is suddenly turned off. With image, Eq. (8.42) reduces to the form of Eq. (8.11). Thus the current must decay exponentially with

image (8.43)

As a more challenging problem, let the circuit be powered by an AC voltage image. Now (8.42) becomes an inhomogeneous equation.

A very useful trick when dealing with quantities having sinusoidal dependence takes advantage of Euler’s theorem (4.57) in the form

image (8.44)

Note that

image (8.45)

Exponentials are much simpler to differentiate and integrate than sines and cosines. Suppose the AC voltage above is imagined to have a complex form image. At the end of the computation, we take the real part of the resulting equation. The answer will be the same, as if we had used image throughout. (A word of caution: make certain that the complexified functions occur only linearly in the equations.)

Accordingly, we write the circuit equation

image (8.46)

where image is a complex variable representing the current. We can separate variables in this equation by introducing the integrating factor image. Since

image (8.47)

Equation (8.46) reduces to

image (8.48)

Integration gives

image (8.49)

If we specify that image when image, we find

image (8.50)

The physically significant result is the real part:

image (8.51)

The last term represents a transient current, which damps out for image. The terms which persist represent the steady-state solution. Note that in the DC limit, as image, the solution reduces to Ohm’s law image. The steady-state solution can be rearranged to the form

image (8.52)

Here the impedance of the RL circuit is defined by image while the phase shift image is given by image.

The steady-state solution can be deduced directly from Eq. (8.46) by assuming from the outset that image. The equation can then be readily solved to give

image (8.53)

which is equivalent to (8.50). One can also define a complex impedance image, in terms of which we can write a complex generalization of Ohm’s law:

image (8.54)

Circuits with image, image, and image involve second-order differential equations, which is our next topic.

8.4 Second-Order Differential Equations

We will consider here linear second-order equations with constant coefficients, in which the functions image and image in Eq. (8.2) are constants. The more general case gives rise to special functions, several of which we will encounter later as solutions of partial differential equations. The homogeneous equation, with image, can be written

image (8.55)

It is convenient to define the differential operator

image (8.56)

in terms of which

image (8.57)

The differential equation (8.55) is then written

image (8.58)

or, in factored form,

image (8.59)

where image are the roots of the auxilliary equation

image (8.60)

The solutions of the two first-order equations

image (8.61)

give

image (8.62)

while

image (8.63)

gives

image (8.64)

Clearly, these are also solutions to Eq. (8.59). The general solution is the linear combination

image (8.65)

In the case that image, one solution is apparently lost. We can recover a second solution by considering the limit:

image (8.66)

(Remember that the partial derivative image does the same thing as image, with every other variable held constant.) Thus the general solution for this case becomes

image (8.67)

When image and image are imaginary numbers, say image and image, the solution (8.65) contains complex exponentials. Since, by Euler’s theorem, these can be expressed as sums and difference of sine and cosine, we can write

image (8.68)

Many applications in physics, chemistry, and engineering involve a simple differential equation, either

image (8.69)

The first equation has trigonometric solutions image and image, while the second has exponential solutions image and image. These results can be easily verified by “reverse engineering.” For example, assuming that image, then image and image. It follows that image.

8.5 Some Examples from Physics

Newton’s second law of motion in one dimension has the form

image (8.70)

where image is the force on a particle of mass image, image is the acceleration, and image is the velocity. Newton’s law leads to a second-order differential equation, with the solution image determining the motion of the particle. The simplest case is a free particle, with image. Newton’s law then reduces to

image (8.71)

The solution is

image (8.72)

in which image and image are the two constants of integration, representing, respectively, the initial image position and velocity. Uniform linear motion at constant velocity image is also in accord with Newton’s first law of motion: a body in motion tends to remain in motion at constant velocity, unless acted upon by some external force.

A slightly more general problem is motion under a constant force. An example is a body in free fall in the general vicinity of the Earth’s surface, which is subject to a constant downward force image. Here image (about 32 ft/s2), the acceleration of gravity. Denoting the altitude above the Earth’s surface by image, we obtain the differential equation

image (8.73)

The factors image cancel. (This is actually a very profound result called the equivalence principle: the gravitational mass of a body equals its inertial mass. It is the starting point for Einstein’s General Theory of Relativity.) One integration of Eq. (8.73) gives

image (8.74)

where the constant of integration image represents the initial velocity. A second integration gives

image (8.75)

where the second constant of integration image represents the initial altitude. This solution is consistent with the well-known result that a falling body (neglecting air resistance) goes about 16 ft in 1 s, 64 ft after 2 s, and so on.

The force exerted by a metal spring subject to moderate stretching or compression is given approximately by Hooke’s law

image (8.76)

Here image represents the displacement, positive or negative, from the spring’s equilibrium extension, as shown in Figure 8.3. The force constantimage is a measure of the spring’s stiffness. The minus sign reflects the fact that the force is in the opposite direction to the displacement—a spring will resist equally either stretching and compression. Consider now a mass image connected to a spring with force constant image. Assume for simplicity that the mass of the spring itself is negligible compared to image. Newton’s second law for this idealized system leads to the differential equation

image (8.77)

The auxilliary equation (8.60) is

image (8.78)

with roots

image (8.79)

Thus the solution is a linear combination of complex exponentials

image (8.80)

Alternatively,

image (8.81)

This shows that the spring oscillates sinusoidally with a natural frequencyimage. Sinusoidal oscillation is also called harmonic motion and the idealized system is referred to as a harmonic oscillator. If necessary, the two constants of integration can be determined by the initial displacement and velocity of the oscillating mass.

image

Figure 8.3 Hooke’s law image for a spring.

Another form for the solution (8.81) can be obtained by setting image. We find then

image (8.82)

An alternative possibility is image.

The oscillation of a real spring will eventually be damped out, in the absence of external driving forces. A reasonable approximation for damping is a force retarding the motion which is proportional to the instantaneous velocity: image, where image is called the damping constant. The differential equation for a damped harmonic oscillator can be written

image (8.83)

where image. The auxilliary equation has the roots image so that the general solution can be written:

image (8.84)

For cases in which damping is not too extreme, so that image, the square roots are imaginary. The solution (8.84) can be written in the form

image (8.85)

This represents a damped sinusoidal wave, as shown in Figure 8.4. For stronger damping, such that image, image decreases exponentially with no oscillation.

image

Figure 8.4 Damped sinusoidal wave of form image. The dotted lines represent the envelopes image.

A damped Hooke’s-law system subject to forced oscillations at frequency image can be represented by an inhomogeneous differential equation:

image (8.86)

Again it is useful to assume a complex exponential driving force image. If we seek, as before, just the steady-state solutions to (8.86), the complex form image can be substituted for image. This reduces (8.86) to an algebraic equation for image:

image (8.87)

with

image (8.88)

The steady-state solution is then given by

image (8.89)

or, more compactly,

image (8.90)

where

image (8.91)

It can be seen that, after transients die out, the forced oscillation impresses its frequency image on the system, apart from a phase shift image. If this frequency matches the natural frequency of the oscillator, image, the system is said to be in resonance. The amplitude of the oscillation then reaches a maximum. At resonance, image, so that the oscillator’s motion is image out of phase with the forcing function.

The behavior of an RLC circuit (Figure 8.2) is closely analogous to that of an oscillating spring. The circuit equation (8.42) generalizes to

image (8.92)

Taking the time derivative leads to a more useful form:

image (8.93)

recalling that image. To obtain the steady-state solution, we assume the complex forms image and image. Equation (8.93) then reduces to an algebraic equation

image (8.94)

The result is most compactly expressed in terms of the impedance:

image (8.95)

Here image is the reactance of the circuit, equal to the difference between inductive and capacitive reactance:

image (8.96)

In terms of the complex impedance

image (8.97)

Equation (8.94) can be solved for

image (8.98)

Therefore the physical value of the current is given by

image (8.99)

The resonance frequency for an RLC circuit is determined by the condition

image (8.100)

At resonance, image, image, and image. The inductive and capacitive reactances exactly cancel so that the impedance reduces to a pure resistance. Thus the current is maximized and oscillates in phase with the voltage. In a circuit designed to detect electromagnetic waves (e.g. radio or television signals) of a given frequency, the inductance and capacitance are “tuned” to satisfy the appropriate resonance condition.

8.6 Boundary Conditions

Often, the boundary conditions imposed on a differential equation determine significant aspects of its solutions. We consider two examples involving the one-dimensional Schrödinger equation in quantum mechanics:

image (8.101)

It will turn out that boundary conditions determine the values of image, the allowed energy levels for a quantum system. A particle-in-a-box is a hypothetical system with a potential energy given by

image (8.102)

Because of the infinite potential nergy, the wavefunction image outside the box, where image or image. This provides two boundary conditions

image (8.103)

for the Schrödinger eqution inside the box, which can be written in the familiar form

image (8.104)

where

image (8.105)

The general solution to (8.104) can be written

image (8.106)

Imposing the boundary condition at image

image (8.107)

which reduces the general solution to

image (8.108)

The boundary condition at image implies

image (8.109)

We can’t just set image because that would imply image everywhere. Recall, however, that the sine function periodically goes through 0, when its argument equals image Therefore the second boundary condition can satisfied if

image (8.110)

This implies that image and, by (8.105), the allowed values of the energy are then given by

image (8.111)

where the integer image is called a quantum number. Quantization of energy in a bound quantum system is thus shown to be a consequence of boundary conditions imposed on the Schrödinger equation. The wavefunction corresponding to the energy image is given by

image (8.112)

Setting image, we obtain wavefunctions fulfilling the normalization condition:

image (8.113)

Problem 8.6.1

Redo the analysis of the same problem with the alternative boundary conditions:

image

For a free particle in quantum mechanics, image everywhere. The Schrödinger equation (8.104) still applies, but now with no restrictive boundary conditions. Any value of image is allowed image and thus image. There is no quantization of energy for a free particle. The wavefunction is conventionally written

image (8.114)

with image [image] corresponding to a particle moving to the right [left]. It is convenient, for bookkeeping purposes to impose periodic boundary conditions, such that

image (8.115)

This requires that

image (8.116)

which is satisfied if

image (8.117)

This implies an artificial quantization of image with

image (8.118)

But, since L can be arbitrarily chosen, all values of image are allowed. With the constant in Eq. (8.114) set equal to image, the wavefunction obeys the box normalization condition

image (8.119)

More generally, since the functions image and image are orthogonal for image, we can write

image (8.120)

in terms of the Kronecker delta (7.78). The functions image thus constitute an orthonormal set.

8.7 Series Solutions

A very general method for obtaining solutions to second-order differential equations is to expand image in a power series and then evaluate the coefficients term by term. We will illustrate the method with a trivial example which we have already solved, namely the equation with constant coefficients:

image (8.121)

Assume that image can be expanded in a power series about image:

image (8.122)

The first derivative is given by

image (8.123)

We have redefined the summation index in order to retain the dependence on image. Analogously,

image (8.124)

Equation (8.121) then implies

image (8.125)

Since this is true for all values of image, every quantity in square brackets must equal zero. This leads to the recursion relation

image (8.126)

Let image be treated as one constant of integration. We then find

image (8.127)

It is convenient to rewrite the coefficient image as image. We find thereby

image (8.128)

The general power-series solution of the differential equation is thus given by

image (8.129)

which is recognized as the expansion for

image (8.130)

We consider next the more general case of a linear homogeneous second-order differential equation with nonconstant coefficients:

image (8.131)

If the functions image and image are both finite at image, then image is called a regular point of the differential equation. If either image or image diverges as image, then image is called a singular point. If both image and image have finite limits as image, then image is called a regular singular point or nonessential singularity. If either of these limits continues to diverge, the point is an essential singularity.

For regular singular points, a series solution of the differential equation can be obtained by the method of Frobenius. This is based on the following generalization of the power-series expansion:

image (8.132)

The derivatives are then given by

image (8.133)

and

image (8.134)

The possible values of image are obtained from the indicial equation, which is based on the presumption that image is the first nonzero coefficient in the series (8.132).

8.8 Bessel Functions

In later work we will encounter Bessel’s differential equation:

image (8.135)

one of the classics of mathematical physics. Bessel’s equation occurs, in particular, in a number of applications involving cylindrical coordinates. Dividing the standard form (8.135) by image shows that image is a regular singular point of Bessel’s equation. The method of Frobenius is thus applicable. Substituting the power-series expansion (8.132) into (8.135), we obtain

image (8.136)

This leads to the recursion relation

image (8.137)

Setting image in the recursion relation and noting that image (image is the first nonvanishing coefficient), we obtain the indicial equation

image (8.138)

The roots are image. With the choice image, the recursion relation simplifies to

image (8.139)

Since image, image (assuming image). Likewise image, as do all odd image. For even image, we have

image (8.140)

For image, the coefficients can be represented by

image (8.141)

From now on, we will use the compact notation

image (8.142)

Setting image, we obtain the conventional definition of a Bessel function of the first kind:

image (8.143)

The first three Bessel functions are plotted in Figure 8.5. Their general behavior can be characterized as damped oscillation, qualitatively similar to that in Figure 8.4.

image

Figure 8.5 Bessel functions of the first kind image.

Bessel functions can be generalized for noninteger index image, as follows:

image (8.144)

The general solution of Bessel’s differential equation for noninteger image is given by

image (8.145)

For integer image, however,

image (8.146)

so that image is not a linearly independent solution. Following the strategy used in Eq. (8.158), we can construct a second solution by defining

image (8.147)

In the limit as image approaches an integer image, we obtain

image (8.148)

This defines a Bessel function of the second kind (sometimes called a Neumann function and written image). The computational details are horrible, but fortunately, mathematicians have worked them all out for us and these functions have been extensively tabulated. Figure 8.6 shows the first three functions image.

image

Figure 8.6 Bessel functions of the second kind image.

The limiting behavior of the image as image is apparent from the leading term image. Using the definition (8.148) we find

image (8.149)

Figure 8.6 shows the logarithmic singularities as image.

8.9 Second Solution

We have encountered two cases in which one solution of a second-order differential equation is relatively straightforward, but the second solution is more obscure. There is a systematic procedure for determining the second solution once the first is known.

Recall that two functions image and image are linearly independent if the relation

image (8.150)

can only be fulfilled when image. A test for linear independence is that the Wronskian, image, of the two functions is not equal to zero:

image (8.151)

If we did have image, then image, which would imply that image, thus negating linear independence.

Assume that image and image are linearly independent solutions to the second-order differential equation (8.131). We can then write

image (8.152)

Multiplying the first equation by image, the second by image and subtracting, we find

image (8.153)

The second bracket is recognized as the Wronskian image, while the first bracket equals image. Thus

image (8.154)

Separation of variables gives

image (8.155)

Thus

image (8.156)

But

image (8.157)

which can be solved for image to give

image (8.158)

For example, suppose we know one solution image for image. Noting that image, we can find the second solution

image (8.159)

Again, for the differential equation image or, more explicitly,

image (8.160)

there is one obvious solution, image. Noting that image, the second solution follows from

image (8.161)

The second solution of Bessel’s equation (8.135) for integer image can be found from (8.158) with image:

image (8.162)

An expansion for image can be obtained after calculating a power series for image.

8.10 Eigenvalue Problems

An operator, designated here by image, represents an action which transforms one function into another function. For the case of functions of a single variable:

image (8.163)

We have already encountered the differential operator image in Section 8.3. In a sense, an operator is a generalization of the concept of a function, which transforms one number into another. For the Schrödinger equation in Eq. (8.101), one can define the Hamiltonian operator by

image (8.164)

The Schrödinger equation can then be written more compactly as

image (8.165)

where E is the energy of the quantum system. As we have seen in Section 8.6, the boundary conditions of the problem determine a set of allowed energies image and corresponding wavefunctions image. These are known as eigenvalues and eigenfunctions, respectively. In pure English these are called “characteristic values” and “characteristic functions,” respectively. The corresponding German terms are eigenwert and eigenfunktion. Current usage employs a hybrid of the English and German words, namely eigenvalue and eigenfunction.

All operators in quantum mechanics which correspond to observable quantities must be Hermitian, such that

image (8.166)

where image and image obey the same analyticity and boundary conditions that are imposed on the eigenfunctions.

The allowed set of eigenvalues and eigenfunctions obtained for some Schrödingier equation can be summarized symbolically by

image (8.167)

For another solution, we can write

image (8.168)

Now multiply the first equation by image and the complex conjugate of the second equation by image. Then subtract the two expressions and integrate over image. The result can be written

image (8.169)

But by the presumed Hermitian property of H, the left-hand side equals zero. Thus

image (8.170)

If image, the second factor becomes image, which is nonzero. Therefore the first factor must equal zero, meaning that

image (8.171)

Thus the energy eigenvalues must be real numbers, which is reasonable, given that they are measurable physical quantities.

Next consider the case when image. Then it must be the second factor in Eq. (8.170) that equals zero:

image (8.172)

Thus eigenfunctions belonging to unequal eigenvalues are orthogonal. There remains the case that for some image, image. The eigenfunctions image and image are then said to be degenerate. But it is always possible to construct linear combinations of degenerate eigenfunctions so that the orthogonality relation still applies. If the eigenfunctions are, in addition, all normalized, we obtain the compact orthonormality condition

image (8.173)

The eigenfunctions image then constitute an orthonormal set.

Problem 8.10.1

Show that a linear combination of degenerate eigenfunctions is itself an eigenfunction with the same energy eigenvalue.


1For interpretation of color in Figure 8.1, the reader is referred to the web version of this book.

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