4. Solving the Schrödinger Equation
Quantum mechanics was developed along two parallel paths, which have come to called the "matrix mechanics" of Werner Heisenberg and the "wave mechanics" of Erwin Schrödinger, though the relationship may not be obvious. And although those two approaches yields equivalent results, each offers benefits in elucidating certain aspects of quantum theory. To help us to understand the connection between matrix mechanics and wave mechanics, the first section explains the meaning of the solutions to the Schrodinger equation using the Born rule, which is the basis for the Copenhagen interpretation of quantum mechanics.
4.1 The Born Rule and Copenhagen Interpretation
When Schrödinger published his equation in 1926, no one (including himself) knew with certainty what the wave equation 𝜓 represented. But Max Born published a paper later in the same year where he explained the only possible interpretation of wavefunction solution to the Schrodinger equation. The "Born rule" says that the quantum wavefunction represents a "probability amplitude" whose magnitude squared determines the probability that a certain result will be obtained when an observation is made.
The most widely accepted (and widely disputed) explanation of quantum mechanics is called "Copenhagen interpretation," since it was developed in large part at Niels Bohr Institute in Copenhagen. Here's a short of each of the principles of Copenhagen interpretation:
Information content The quantum state 𝛹 include all possible information about a quantum system - there are no "hidden variables" with additional information.
Time evolution Over time, quantum states evolve smoothly in accordance with the Schrödinger equation unless a measurement is made.
Wavefunction collapse Whenever a measurement of a quantum state is made, the states "collapse" to an eigenstate of the operator associated with the observable being measured.
Measurement results The value measured for an observable is the eigenvalue of the eigenstates to which the original quantum states has collapsed.
Uncertainty principle Certain "incompatible" observable (such as position and momentum) may not be simultaneously known with arbitrarily great precision.
Born rule The probability that a quantum state will collapse to a given eigenstate upon measurement is determined by the square of the amount of that eigenstate present in the original state (the wavefunction)
Correspondence principle In the limit of very large quantum numbers, the results of measurements of quantum observables must match the results of classical physics.
Complementarity Every quantum system includes complementary wave-like and particle-like aspects; whether the system behaves like a wave or like a particle when measured is determined by the nature of the measurement.
The "mechanics" of quantum mechanics works. That is, the quantum mechanical observables and calculating the probability of each of those outcomes has been repeatedly demonstrated to give correct answers.
4.2 Quantum States, Wavefunctions, and Operators
Fundamentally probabilistic nature of quantum mechanics leads to several profound differences with classical mechanics.
The "quantum state" or a particle or system is a description that contains all the information that can be known about the particle or system. A quantum states is usually written as 𝜓 (sometimes uppercase 𝛹, especially when time dependence is included) and can be represented bu a basis-independent ket ∣𝜓⟩ or ∣𝛹⟩. Quantum states are members of an abstract vector space, and the Schrodinger equation describes how a quantum states evolves over time.
A "quantum wavefunction" is defined as the expansion of a quantum state in a specified basis. Whichever basis we can choose, and a logical choice is the basis corresponding to the observable of interest. Recall that every observable is associated with an operator, and the eigenfunctions of that operator form a complete orthogonal basis. That means that any function may be synthesized by weighted combination (superposition) of those eigenfunctions. So we can expand quantum state using a weighted sum of the eigenfunction (𝜓1, 𝜓2, ∙ ∙ ∙ , 𝜓𝑁) for that basis, then a state ∣𝜓⟩ may be written as
(1.35) ∣𝜓⟩ = 𝑐1∣𝜓1⟩ + 𝑐2∣𝜓2⟩ + ∙ ∙ ∙ + 𝑐𝑁∣𝜓𝑁⟩ = ∑𝑛=1𝑁 𝑐𝑛∣𝜓𝑛⟩
and the wavefunction is the amount (𝑐𝑛) of each eigenfunction ∣𝜓𝑛⟩ in state ∣𝜓⟩. So the wavefunction in a specified basis is the collection of (potentially complex) values 𝑐𝑛 for that basis. And the each 𝑐𝑛 may be found by projecting state onto corresponding (normalized) eigenfunction ∣𝜓𝑛⟩:
(4.1) 𝑐𝑛 = ⟨𝜓𝑛∣𝜓⟩.
In quantum textbooks, the word "function" which is applied to continuous values 𝑐𝑛 is sometimes described as the operator having a "continuous spectrum" of eigenvalues.
For example, the 1-D position basis function may be represented by the ket ∣𝑥⟩, so the expanding the state represented in the position basis looks like this:
(4.2) ∣𝜓⟩ = ∫-∞∞ 𝜓(𝑥) ∣𝑥⟩ 𝑑𝑥.
To determine 𝜓(𝑥) let us project the state ∣𝜓⟩ onto the position basis functions:
(4.3) 𝜓(𝑥) = ⟨𝑥∣𝜓⟩.
Just as in the discrete case, ∣𝜓(𝑥)∣2 gives us the probability density (the probability per unit length in the 1-D case), which we must integrate over a range of 𝑥 to determine the probability of an outcome within that range.
The same approach can be taken for momentum wavefunction. The 1-D momentum basis functions may be represented by the ket ∣𝑝⟩, and the expanding the case the state ∣𝜓⟩ in the momentum basis looks like this:
(4.4) ∣𝜓⟩ = ∫-∞∞ 𝜙̄(𝑝) ∣𝑝⟩ 𝑑𝑝.
To determine 𝜙̄(𝑝), let us project the state ∣𝜓⟩ onto the momentum basis functions:
(4.5) 𝜙̄(𝑝) = ⟨𝑝∣𝜓⟩.
It is worth to noting that to determine the position or momentum waveform is to project a given quantum state ∣𝜓⟩ onto the eigenstates using the inner product, and the position-basis and momentum-basis wavefunctions related to one another by the Fourier transform, not by using the position or momentum operator.
Also since operating on a state does produce a new state, that is, making a measurement causes the collapse of a quantum wavefunction, the actual relationship between applying an operator and making a measurement is a bit more complex, and more informative. That is, after applying the operator to the quantum state, the weighting factor for each eigenstate includes the eigenvalue of that eigenstates, because the operator has brought out a factor of that eigenvalue (𝑜𝑛):
(4.6) Ô ∣𝜓⟩ = ∑𝑛𝑐𝑛Ô ∣𝜓⟩ = ∑𝑛𝑐𝑛𝑜𝑛∣𝜓𝑛⟩.
As explained in Section 2.5, forming the inner product of this new state Ô ∣𝜓⟩ with the original state ∣𝜓⟩ gives the expectation value of the observable corresponding to the operator. So in the case of observable 𝑂 with associated operator Ô. eigenvalue 𝑜𝑛 and expectation value ⟨𝑂⟩,
⟨𝜓∣ Ô ∣𝜓⟩ = ∑𝑚(𝑐*𝑚∣𝜓𝑚) ∑𝑛(𝑐𝑛𝑜𝑛∣𝜓𝑛⟩) = ∑𝑚∑𝑛(𝑐*𝑚𝑜𝑛𝑐𝑛) ⟨𝜓𝑚∣𝜓𝑛⟩ = ∑𝑛𝑜𝑛(∣𝑐𝑛∣)2 = ⟨𝑂⟩,
since ⟨𝜓𝑚∣𝜓𝑛⟩ = 𝛿𝑚,𝑛 for orthogonal wavefunctions.
We must note the role of operator: it has performed the function of producing a new state in which the weighting coefficient of each eigenfunction has been multiplied by its the eigenvalue. And making a measurement changes the quantum state by causing the wavefunction to collapse to one of those eigenfunctions. So operator and measurements both change the quantum state of a system.
4.3 Characteristics of Quantum Wavefunctions
In order to determine the details of the quantum wavefunctions that are solutions to the Schrödinger equation, we need to know the value of the potential energy 𝑉 over the region of interest.
For a function to qualify as a quantum wavefunction, it must be a solution to the Schrödinger equation, and it must also meet the requirements of the Born rule relating the squared magnitude of the function to the probability or probability density. Such functions are described as "well-behaved", that is, the functions must be single-valued, smooth, and square-integrable as follows:
Single-valued This means that the wavefunction can have only one value, because the Born rule tells us that the square of the wavefunction gives the probability which can have only one value at any location.
Smooth This means that the wavefunction and its first spatial derivative must be continuous. That's because the Schrödinger equation is a second-order differential equation in the spatial coordinate. An exception to this occurs in the case of infinite potential.
Square-intergrable Quantum wavefunctions must be normalizable. That means the function itself must be finite everywhere but the Dirac delta function is an exception. Although the the Dirac delta function is defined to have infinite height, its infinitely narrow width keeps the area under its curve finite. Also some functions such as the plane-wave function 𝐴𝑒𝑖(𝑘𝑥-𝜔𝑡) have infinite spatial extent and are not individually square-intergrable, but it is possible to make them meet the requirement of square-integrability.
In addition, quantum wavefunction must also match the boundary conditions for a particular problem. Let consider the time-independent Schrödinger equation(TISE). To understand that behavior, we rearrange the TISE(Eq. 3.40):
(4.7) 𝑑2[𝜓(𝑥)]/𝑑𝑥2 = -2𝑚/ℏ2 (𝐸 - 𝑉)𝜓(𝑥).
The term on the left side is just the spatial curvature, which is the change in the slope of a graph of the 𝜓(𝑥 vs. location.
Now imagine a situation in which the total energy is greater than potential energy in the region of interest. hence the 𝐸 - 𝑉 is positive, if 𝜓(𝑥) > 0, curvature is negative (slope becomes less positive or more negative as 𝑥 gets larger and if if 𝜓(𝑥) < 0, curvature is positive.
This means that when 𝜓(𝑥) > 0, the graph of 𝜓(𝑥) becomes increasingly negative as 𝑥 moves to larger value, the waveform must curve toward the x-axis, eventually crossing that axis. Then 𝜓(𝑥) becomes negative and the waveform again curves toward the x-axis, until it eventually cross back the positive-𝜓 region. So as long as total energy exceeds the potential energy, the wavefunction 𝜓(𝑥) will oscillate as a function of position.
Now consider a region in which the total energy is less than the potential energy, that is, 𝐸 - 𝑉 is negative, so the curvature has the same sign of the wavefunction. In the classical physics, the regions are called "classically forbidden" or "classically disallowed". Since position and momentum are incompatible observables, and kinetic energy depends on the square of momentum, the uncertainty principle says that we cannot simultaneously measure both position and kinetic energy with arbitrarily great precision. So when we measure kinetic energy and get a positive value, the possible positions of quantum particle or system always include the classically forbidden region.
With that understanding, let take a look at the behavior of 𝜓(𝑥) as we move toward positive 𝑥 in which 𝜓(𝑥) is initially positive in Fig. 4.3. In the figure, the upper dot line shows the case that 𝜓(𝑥) has the initially slope positive, the middle dot line shows the case that initial slope is slightly negative, and the lower one shows the case that its initial slope are very negative separately. But since wavefunctions with infinitely large amplitude are not physically realizable, its curvature must be such that the amplitude of 𝜓(𝑥) remains finite at all location so that the wavefunction is normalizable. This means that the value of 𝜓(𝑥) must tend toward zero as 𝑥 approaches ∓∞, never turning away from the axis, but also never crossing below the axis as shown in Fig. 4.3. In conclusion oscillations are not possible in the region in which 𝐸 - 𝑉 is negative.
To get a batter sense of how the wavefunction behave first taking the case in which 𝑉 is constant and 𝐸 - 𝑉 is positive in the region of interest. the TISE(Eq. 4.7) can be written as
(4.8) 𝑑2[𝜓(𝑥)]/𝑑𝑥2 = -2𝑚/ℏ2 (𝐸 - 𝑉)𝜓(𝑥) = -𝑘2𝜓(𝑥).
in which the constant 𝑘 is given by
(4.9) 𝑘 = √[2𝑚/ℏ2 (𝐸 - 𝑉)].
We can find the general solution to this equation (using WolframAlpa conveniently)
(4.10) 𝜓(𝑥) = 𝛢𝑒𝑖𝑘𝑥 + 𝐵𝑒-𝑖𝑘𝑥,
where 𝛢 and 𝐵 are constant to be determined by the boundary conditions.
Even without knowing those boundary conditions, we can see 𝜓(𝑥) oscillate sinusoidally in the region, since 𝑒∓𝑖𝑘𝑥 = cos 𝑘𝑥 ∓ sin 𝑘𝑥 by Euler's relation. And in Eq. 4.10, 𝑘 represents the wavenumber with the relation 𝑘 = 2π/𝜆. The wavenumber determines how "fast" 𝜓(𝑥) oscillates with distance. If we analyze Eq. 4.10 further, we may conclude that the ratio of amplitudes on opposite sides of the boundary is inversely proportional to the wavenumber ratio. The larger energy difference 𝐸 - 𝑉 is, the smaller amplitude the wavefunction must have than the opposite side of the boundary.
Now let consider the case 𝐸 - 𝑉 is negative and then the write TISE(Eq. 4.7), 𝜅 and its general solution as follows
(4.11) 𝑑2[𝜓(𝑥)]/𝑑𝑥2 = -2𝑚/ℏ2 (𝐸 - 𝑉)𝜓(𝑥) = +𝜅2𝜓(𝑥),
(4.12) 𝜅 = √[2𝑚/ℏ2 (𝑉 - 𝐸)],
(4.13) 𝜓(𝑥) = 𝐶𝑒𝜅𝑥 + 𝐷𝑒-𝜅𝑥
where 𝐶 and 𝐷 are constant to be determined by the boundary conditions.
In this region, when 𝑥 goes toward +∞ the first term will become infinitely large unless the coefficient 𝐶 is set to zero. So in this region, 𝜓(𝑥) = 𝐷𝑒-𝜅𝑥, which decrease exponentially with increasing positive 𝑥. Likewise, when 𝑥 goes toward -∞ the first term will become infinitely large unless the coefficient 𝐷 is set to zero and so 𝜓(𝑥) = 𝐶𝑒𝜅𝑥, which decrease exponentially with increasing positive 𝑥.
So once again, in this region, without the precise boundary conditions, we can conclude that quantum wavefunctions decay exponentially accordance with the curvature analysis presented earlier.
We can see all of these characteristics at work in Fig. 4.4, which shoes five regions with different potential values 𝑉(𝑥) which is constant within each region,
In classically forbidden regions 1 and 5 the wavefunctions decays exponentially and since 𝑉 - 𝐸 is greater in region 5 than in region 1, the decay over distance is faster.
In classically allowed region 2, the total energy and the potential energy are equal, so the curvature is zero in that region. And in the region 3 and 4, the wavefunction oscillates. The larger value of 𝐸 - 𝑉 in the region 4 makes for shorter wavelength and smaller amplitude in that region.
It is worth to noting that for a particle with the total energy 𝐸 shown in the figure, the probability of finding the particle decreases to zero as 𝑥 approaches ∓∞.
4.4.1 Fourier Theory: Analysis and Synthesis
The goal of this section is to understand why the Fourier transform plays a key role in both analysis and synthesis, and exactly how that transform works. And later we will see how Fourier theory can be used to understand quantum wave packets and how it relates to the uncertainty principle.
Let consider the mathematical statement of the Fourier transform of a function of position 𝜓(𝑥):
(4.14) 𝜙(𝑘) = 1/√2π ∫-∞∞𝜓(𝑥)𝑒-𝑖𝑘𝑥 𝑑𝑥,
where 𝜙(𝑘) is a function of wavenumber 𝑘, called the wavenumber spectrum. If we already know 𝜙(𝑘) and we want to determine the corresponding position function, the tool we need is the inverse Fourier transform
(4.15) 𝜓(𝑥) = 1/√2π ∫-∞∞𝜙(𝑘)𝑒𝑖𝑘𝑥 𝑑𝑘,
Forier theory is rooted in one idea: any well behaved function can be expressed as a weighted combination of sinusoidal functions. Eq. 4.14 is telling us is this: to find those amounts, multiply 𝜓(𝑥) by 𝑒-𝑖𝑘𝑥 (which is is equivalent to cos 𝑘𝑥 - 𝑖sin 𝑘𝑥 by Euler's relation) and integrate the resulting product over all space.
There are several ways to picture this, but one may find that the simplest visualization comes from using Euler's relation to write the Fourier transformation as
(4.16) 𝜙(𝑘) = 1/√2π ∫-∞∞𝜓(𝑥)𝑒-𝑖𝑘𝑥 𝑑𝑥 = 1/√2π ∫-∞∞𝜓(𝑥)cos (𝑘𝑥) 𝑑𝑥 - 𝑖(1/√2π) ∫-∞∞𝜓(𝑥)sin (𝑘𝑥) 𝑑𝑥.
Now imagine the case 𝜓(𝑥) = cos (𝑘1𝑥), then we get
𝜙(𝑘) = 1/√2π ∫-∞∞cos (𝑘1𝑥) cos (𝑘𝑥) 𝑑𝑥 - 𝑖(1/√2π) ∫-∞∞cos (𝑘1𝑥) sin (𝑘𝑥) 𝑑𝑥.
The next step is to invoke the "orthogonality relations", which say that the first integral is nonzero only when 𝑘 = 𝑘1, while the integral is zero for all values of 𝑘. So in the case 𝑘 = 𝑘1, as we can see in Fig. 4.5, the multiplication results are all positive albeit with varying amplitude due to the oscillations of both functions. Integrating the multiplication product over 𝑥 is equivalent to find the under this curve. So if the integration extends from -∞ to +∞, the area under curve is infinitive. But even the slightest difference between 𝑘 and 𝑘1, will cause the product of cos 𝑘𝑥 and cos 𝑘1𝑥 to oscillate between positive and negative values. That means that the result of integration over all space approaches infinity for 𝑘 = 𝑘1 and zero for all other values of 𝑘, resulting in a function that's infinitely tall but infinitely narrow. That function is the Dirac delta function.
Since 𝜓(𝑥) is pure cosine wave, the imaginary portion of 𝜙(𝑘), even at the wavenumber 𝑘1, if we integrate over an integer number of cycles of 𝜓(𝑥), the Fourier transform produce precisely zero. And if 𝑘 ≠ 𝑘1, the area under the resulting curve tends towards zero.
When 𝜓(𝑥) is pure sine wave, the result 𝜙(𝑘) of The Fourier transform process would have been purely imaginary. In general, the result of the Fourier transform is complex, whether the function is purely real, purely imaginary, or complex.
There is another way is to represent the components of 𝜓(𝑥) and the multiplying factor 𝑒-𝑖𝑘𝑥 as phasors. A phasor is a type of vector in 2-D space defined by a "real" axis and a perpendicular "imaginary" axis, typically drawn with its base at the origin and its tip on the unit circle, which is the locus of the points at a distance of one unit from the origin. In Fog. 4.9, we can see the function cos (𝑘𝑥) and sin (𝑘𝑥) are traced out by projecting the phasor onto the real and imaginary axes and eight randomly selected values of the angle 𝑘𝑥. Since 𝑘 = 2π/𝜆, the phasor will make one complete revolution by 2π radians.
To use phasors to understand the Fourier transform, imagine that 𝜓(𝑥) = 𝑒𝑖𝑘1𝑥 and in Fig. 4.10a 𝜓(𝑥) is shown at 10 evenly spaced values of 𝑘1𝑥. Now look at Fig. 4.10b, which shows the phasor representing multiplying factor 𝑒-𝑖𝑘1𝑥 for the case 𝑘 = 𝑘1. The phasor rotates at the same rate as 𝜓(𝑥) but the negative sign in the exponent means that it rotates clockwise. So multiplying two phasors produces another phasor, and the amplitude of that new phasor is equal to the product of the amplitudes of the two phasors. More importantly for this implication, the direction of the new phasor is equal to the sum of the angles of the two phasors.※ Hence the multiplications performed at the ten phasor angles all the result in a unit-length phasor pointing along the real axis, as shown in Fig. 4.10c.
Because the Fourier-transform process integrates the result of that multiplication over all values 𝑥:
(4.14) 𝜙(𝑘) = 1/√2π ∫-∞∞𝜓(𝑥)𝑒-𝑖𝑘𝑥 𝑑𝑥,
which is equivalent to continuously adding the resulting phasors. So when 𝑘 = 𝑘1, the addition yields a large number and the large number becomes infinitives when the value of 𝑘 precisely equals the values of 𝑘1 and the integration extends from -∞ to ∞.
When 𝑘1 does not match the wavenumber of the multiplying function 𝑒-𝑖𝑘𝑥, the result of integrating the product of them will be exactly zero if the integration is performed over a sufficiently large range of 𝑥 to cause the product phasor to complete an integer number of circles, as we may guess.
Also we can use this same type of phasor analysis for a single rotating phasor with constant amplitude. For example, let consider 𝜓(𝑥) = cos (𝑘1𝑥) and may recall the "inverse Euler" relation for cosine:
(4.17) cos (𝑘1𝑥) = 1/2 (𝑒𝑖𝑘𝑥 + 𝑒-𝑖𝑘𝑥).
This means that 𝜓(𝑥) = cos (𝑘1𝑥) may represented by two counter-rotating phasors with amplitude of 1/2. So for the case 𝑘 = 𝑘1, with the multiplying factor 𝑒-𝑖𝑘𝑥, as expected, the result is real, and its amplitude is given by the sum of the product phasors.※ So the rotating phasors can be helpful in visualizing the process of Fourier transformation.
To understand the connection of Fourier analysis to quantum wavefunctions, it's helpful to express these process using Dirac notation. To do that recall that the position-basis wavefunction 𝜓(𝑥) is the projection of a basis-independent state vector, ket ∣𝜓⟩ onto the position basis vector, ∣𝑥⟩:
(4.18) 𝜓(𝑥) = ⟨𝑥∣𝜓⟩.
Also note that the plane-wave functions 1/√2π 𝑒𝑖𝑘𝑥are the wavenumber eigenfunctions represented in the position basis
(4.19) 1/√2π 𝑒𝑖𝑘𝑥 = ⟨𝑥∣𝜓⟩.
Rearranging the Fourier transform Eq. 4.14 as
𝜙(𝑘) = 1/√2π ∫-∞∞𝜓(𝑥)𝑒-𝑖𝑘𝑥 𝑑𝑥 = ∫-∞∞ 1/√2π 𝑒-𝑖𝑘𝑥 𝜓(𝑥) 𝑑𝑥
and then inserting ⟨𝑥∣𝑘⟩* for 1/√2π 𝑒-𝑖𝑘𝑥 and ⟨𝑥∣𝜓⟩ for 𝜓(𝑥) gives
(4.20) 𝜙(𝑘) = ∫-∞∞ ⟨𝑥∣𝑘⟩* ⟨𝑥∣𝜓⟩ 𝑑𝑥 = ∫-∞∞ ⟨𝑘∣𝑥⟩ ⟨𝑥∣𝜓⟩ 𝑑𝑥 = ⟨𝑘∣ Î ∣𝜓⟩,
where Î represents the identity operator. Since ∣𝑥⟩ ⟨𝑥∣ is a projection operator and the sum of a projection operator over all basis vectors gives the identity operator (refer to Section 2.4).
(4.21) 𝜙(𝑘) = ⟨𝑘∣ Î ∣𝜓⟩ = ⟨𝑘∣𝜓⟩,
where the state ∣𝜓⟩ corresponds to the position-basis wavefunction 𝜓(𝑥), and the wavenumber ket ∣𝑘⟩ corresponds to the plane-wave sinusoidal basis functions 1/√2π 𝑒𝑖𝑘𝑥.
Whether we think that the Fourier transform as multiplying a function by cosine and sine functions and integrating the products, or as multiplying and adding phasors, or as projecting an abstract state vector onto sinusoidal wavenumber basis function, the bottom line is that Fourier analysis provides a means of determining the amount of each of the constituent sinusoidal waves that make up the function.
* Refer to Eq. 1.36 (Section 1.6).
※ attention: some rigorous derivation might be required