Classical mechanics applies only to macroscopic particles. For microscopic "particles" we require a new form of mechanics, called quantum mechanics. We now consider some of the contrasts between classical and quantum mechanics. For simplicity a one-particle, one-dimensional system will be discussed.
In classical mechanics the motion of a particle is governed by Newton's second law:
\(\begin{equation*}F=m a=m \frac{d^{2} x}{d t^{2}} \tag{1.8}\end{equation*}\)
where $F$ is the force acting on the particle, $m$ is its mass, and $t$ is the time; $a$ is the acceleration, given by $a=d v / d t=(d / d t)(d x / d t)=d^{2} x / d t^{2}$, where $v$ is the velocity. Equation (1.8) contains the second derivative of the coordinate $x$ with respect to time. To solve it, we must carry out two integrations. This introduces two arbitrary constants $c_{1}$ and $c_{2}$ into the solution, and
\(\begin{equation*}x=g\left(t, c_{1}, c_{2}\right) \tag{1.9}\end{equation*}\)
where $g$ is some function of time. We now ask: What information must we possess at a given time $t_{0}$ to be able to predict the future motion of the particle? If we know that at $t_{0}$ the particle is at point $x_{0}$, we have
\(\begin{equation*}x_{0}=g\left(t_{0}, c_{1}, c_{2}\right) \tag{1.10}\end{equation*}\)
Since we have two constants to determine, more information is needed. Differentiating (1.9), we have
\(\frac{d x}{d t}=v=\frac{d}{d t} g\left(t, c_{1}, c_{2}\right)\)
If we also know that at time $t_{0}$ the particle has velocity $v_{0}$, then we have the additional relation
\(\begin{equation*}v_{0}=\left.\frac{d}{d t} g\left(t, c_{1}, c_{2}\right)\right|_{t=t_{0}} \tag{1.11}\end{equation*}\)
We may then use (1.10) and (1.11) to solve for $c_{1}$ and $c_{2}$ in terms of $x_{0}$ and $v_{0}$. Knowing $c_{1}$ and $c_{2}$, we can use Eq. (1.9) to predict the exact future motion of the particle.
As an example of Eqs. (1.8) to (1.11), consider the vertical motion of a particle in the earth's gravitational field. Let the $x$ axis point upward. The force on the particle is downward and is $F=-m g$, where $g$ is the gravitational acceleration constant. Newton's second law (1.8) is $-m g=m d^{2} x / d t^{2}$, so $d^{2} x / d t^{2}=-g$. A single integration gives $d x / d t=-g t+c_{1}$. The arbitrary constant $c_{1}$ can be found if we know that at time $t_{0}$ the particle had velocity $v_{0}$. Since $v=d x / d t$, we have $v_{0}=-g t_{0}+c_{1}$ and $c_{1}=v_{0}+g t_{0}$. Therefore, $d x / d t=-g t+g t_{0}+v_{0}$. Integrating a second time, we introduce another arbitrary constant $c_{2}$, which can be evaluated if we know that at time $t_{0}$ the particle had position $x_{0}$. We find (Prob. 1.7) $x=x_{0}-\frac{1}{2} g\left(t-t_{0}\right)^{2}+v_{0}\left(t-t_{0}\right)$. Knowing $x_{0}$ and $v_{0}$ at time $t_{0}$, we can predict the future position of the particle.
The classical-mechanical potential energy $V$ of a particle moving in one dimension is defined to satisfy
\(\begin{equation*}\frac{\partial V(x, t)}{\partial x}=-F(x, t) \tag{1.12}\end{equation*}\)
For example, for a particle moving in the earth's gravitational field, $\partial V / \partial x=-F=m g$ and integration gives $V=m g x+c$, where $c$ is an arbitrary constant. We are free to set the zero level of potential energy wherever we please. Choosing $c=0$, we have $V=m g x$ as the potential-energy function.
The word state in classical mechanics means a specification of the position and velocity of each particle of the system at some instant of time, plus specification of the forces
acting on the particles. According to Newton's second law, given the state of a system at any time, its future state and future motions are exactly determined, as shown by Eqs. (1.9)-(1.11). The impressive success of Newton's laws in explaining planetary motions led many philosophers to use Newton's laws as an argument for philosophical determinism. The mathematician and astronomer Laplace (1749-1827) assumed that the universe consisted of nothing but particles that obeyed Newton's laws. Therefore, given the state of the universe at some instant, the future motion of everything in the universe was completely determined. A super-being able to know the state of the universe at any instant could, in principle, calculate all future motions.
#### Abstract
Although classical mechanics is deterministic, many classical-mechanical systems (for example, a pendulum oscillating under the influence of gravity, friction, and a periodically varying driving force) show chaotic behavior for certain ranges of the systems' parameters. In a chaotic system, the motion is extraordinarily sensitive to the initial values of the particles' positions and velocities and to the forces acting, and two initial states that differ by an experimentally undetectable amount will eventually lead to very different future behavior of the system. Thus, because the accuracy with which one can measure the initial state is limited, prediction of the long-term behavior of a chaotic classical-mechanical system is, in practice, impossible, even though the system obeys deterministic equations. Computer calculations of solar-system planetary orbits over tens of millions of years indicate that the motions of the planets are chaotic [I. Peterson, Newton's Clock: Chaos in the Solar System, Freeman, 1993; J. J. Lissauer, Rev. Mod. Phys., 71, 835 (1999)].
Given exact knowledge of the present state of a classical-mechanical system, we can predict its future state. However, the Heisenberg uncertainty principle shows that we cannot determine simultaneously the exact position and velocity of a microscopic particle, so the very knowledge required by classical mechanics for predicting the future motions of a system cannot be obtained. We must be content in quantum mechanics with something less than complete prediction of the exact future motion.
Our approach to quantum mechanics will be to postulate the basic principles and then use these postulates to deduce experimentally testable consequences such as the energy levels of atoms. To describe the state of a system in quantum mechanics, we postulate the existence of a function $\Psi$ of the particles' coordinates called the state function or wave function (often written as wavefunction). Since the state will, in general, change with time, $\Psi$ is also a function of time. For a one-particle, one-dimensional system, we have $\Psi=\Psi(x, t)$. The wave function contains all possible information about a system, so instead of speaking of "the state described by the wave function $\Psi$," we simply say "the state $\Psi$." Newton's second law tells us how to find the future state of a classicalmechanical system from knowledge of its present state. To find the future state of a quantum-mechanical system from knowledge of its present state, we want an equation that tells us how the wave function changes with time. For a one-particle, one-dimensional system, this equation is postulated to be
\(\begin{equation*}-\frac{\hbar}{i} \frac{\partial \Psi(x, t)}{\partial t}=-\frac{\hbar^{2}}{2 m} \frac{\partial^{2} \Psi(x, t)}{\partial x^{2}}+V(x, t) \Psi(x, t) \tag{1.13}\end{equation*}\)
where the constant $\hbar$ (h-bar) is defined as
\(\begin{equation*}\hbar \equiv \frac{h}{2 \pi} \tag{1.14}\end{equation*}\)
The concept of the wave function and the equation governing its change with time were discovered in 1926 by the Austrian physicist Erwin Schrödinger (1887-1961). In this equation, known as the time-dependent Schrödinger equation (or the Schrödinger wave equation), $i=\sqrt{-1}, m$ is the mass of the particle, and $V(x, t)$ is the potentialenergy function of the system. (Many of the historically important papers in quantum mechanics are available at dieumsnh.qfb.umich.mx/archivoshistoricosmq.)
The time-dependent Schrödinger equation contains the first derivative of the wave function with respect to time and allows us to calculate the future wave function (state) at any time, if we know the wave function at time $t_{0}$.
The wave function contains all the information we can possibly know about the system it describes. What information does $\Psi$ give us about the result of a measurement of the $x$ coordinate of the particle? We cannot expect $\Psi$ to involve the definite specification of position that the state of a classical-mechanical system does. The correct answer to this question was provided by Max Born shortly after Schrödinger discovered the Schrödinger equation. Born postulated that for a one-particle, one-dimensional system,
\(\begin{equation*}|\Psi(x, t)|^{2} d x \tag{1.15}\end{equation*}\)
gives the probability at time $t$ of finding the particle in the region of the $x$ axis lying between $x$ and $x+d x$. In (1.15) the bars denote the absolute value and $d x$ is an infinitesimal length on the $x$ axis. The function $|\Psi(x, t)|^{2}$ is the probability density for finding the particle at various places on the $x$ axis. (Probability is reviewed in Section 1.6.) For example, suppose that at some particular time $t_{0}$ the particle is in a state characterized by the wave function $a e^{-b x^{2}}$, where $a$ and $b$ are real constants. If we measure the particle's position at time $t_{0}$, we might get any value of $x$, because the probability density $a^{2} e^{-2 b x^{2}}$ is nonzero everywhere. Values of $x$ in the region around $x=0$ are more likely to be found than other values, since $|\Psi|^{2}$ is a maximum at the origin in this case.
To relate $|\Psi|^{2}$ to experimental measurements, we would take many identical noninteracting systems, each of which was in the same state $\Psi$. Then the particle's position in each system is measured. If we had $n$ systems and made $n$ measurements, and if $d n_{x}$ denotes the number of measurements for which we found the particle between $x$ and $x+d x$, then $d n_{x} / n$ is the probability for finding the particle between $x$ and $x+d x$. Thus
\(\frac{d n_{x}}{n}=|\Psi|^{2} d x\)
and a graph of $(1 / n) d n_{x} / d x$ versus $x$ gives the probability density $|\Psi|^{2}$ as a function of $x$. It might be thought that we could find the probability-density function by taking one system that was in the state $\Psi$ and repeatedly measuring the particle's position. This procedure is wrong because the process of measurement generally changes the state of a system. We saw an example of this in the discussion of the uncertainty principle (Section 1.3).
Quantum mechanics is statistical in nature. Knowing the state, we cannot predict the result of a position measurement with certainty; we can only predict the probabilities of various possible results. The Bohr theory of the hydrogen atom specified the precise path of the electron and is therefore not a correct quantum-mechanical picture.
Quantum mechanics does not say that an electron is distributed over a large region of space as a wave is distributed. Rather, it is the probability patterns (wave functions) used to describe the electron's motion that behave like waves and satisfy a wave equation.
How the wave function gives us information on other properties besides the position is discussed in later chapters.
The postulates of thermodynamics (the first, second, and third laws of thermodynamics) are stated in terms of macroscopic experience and hence are fairly readily understood. The postulates of quantum mechanics are stated in terms of the microscopic world and appear quite abstract. You should not expect to fully understand the postulates of quantum mechanics at first reading. As we treat various examples, understanding of the postulates will increase.
It may bother the reader that we wrote down the Schrödinger equation without any attempt to prove its plausibility. By using analogies between geometrical optics and classical mechanics on the one hand, and wave optics and quantum mechanics on the other hand, one can show the plausibility of the Schrödinger equation. Geometrical optics is an approximation to wave optics, valid when the wavelength of the light is much less than the size of the apparatus. (Recall its use in treating lenses and mirrors.) Likewise, classical mechanics is an approximation to wave mechanics, valid when the particle's wavelength is much less than the size of the apparatus. One can make a plausible guess as to how to get the proper equation for quantum mechanics from classical mechanics based on the known relation between the equations of geometrical and wave optics. Since many chemists are not particularly familiar with optics, these arguments have been omitted. In any case, such analogies can only make the Schrödinger equation seem plausible. They cannot be used to derive or prove this equation. The Schrödinger equation is a postulate of the theory, to be tested by agreement of its predictions with experiment. (Details of the reasoning that led Schrödinger to his equation are given in Jammer, Section 5.3. A reference with the author's name italicized is listed in the Bibliography.)
Quantum mechanics provides the law of motion for microscopic particles. Experimentally, macroscopic objects obey classical mechanics. Hence for quantum mechanics to be a valid theory, it should reduce to classical mechanics as we make the transition from microscopic to macroscopic particles. Quantum effects are associated with the de Broglie wavelength $\lambda=h / m v$. Since $h$ is very small, the de Broglie wavelength of macroscopic objects is essentially zero. Thus, in the limit $\lambda \rightarrow 0$, we expect the time-dependent Schrödinger equation to reduce to Newton's second law. We can prove this to be so (see Prob. 7.59).
A similar situation holds in the relation between special relativity and classical mechanics. In the limit $v / c \rightarrow 0$, where $c$ is the speed of light, special relativity reduces to classical mechanics. The form of quantum mechanics that we will develop will be nonrelativistic. A complete integration of relativity with quantum mechanics has not been achieved.
Historically, quantum mechanics was first formulated in 1925 by Heisenberg, Born, and Jordan using matrices, several months before Schrödinger's 1926 formulation using differential equations. Schrödinger proved that the Heisenberg formulation (called matrix mechanics) is equivalent to the Schrödinger formulation (called wave mechanics). In 1926, Dirac and Jordan, working independently, formulated quantum mechanics in an abstract version called transformation theory that is a generalization of matrix mechanics and wave mechanics. In 1948, Feynman devised the path integral formulation of quantum mechanics [R. P. Feynman, Rev. Mod. Phys., 20, 367 (1948); R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, 1965].
In classical mechanics the motion of a particle is governed by Newton's second law:
\(\begin{equation*}F=m a=m \frac{d^{2} x}{d t^{2}} \tag{1.8}\end{equation*}\)
where $F$ is the force acting on the particle, $m$ is its mass, and $t$ is the time; $a$ is the acceleration, given by $a=d v / d t=(d / d t)(d x / d t)=d^{2} x / d t^{2}$, where $v$ is the velocity. Equation (1.8) contains the second derivative of the coordinate $x$ with respect to time. To solve it, we must carry out two integrations. This introduces two arbitrary constants $c_{1}$ and $c_{2}$ into the solution, and
\(\begin{equation*}x=g\left(t, c_{1}, c_{2}\right) \tag{1.9}\end{equation*}\)
where $g$ is some function of time. We now ask: What information must we possess at a given time $t_{0}$ to be able to predict the future motion of the particle? If we know that at $t_{0}$ the particle is at point $x_{0}$, we have
\(\begin{equation*}x_{0}=g\left(t_{0}, c_{1}, c_{2}\right) \tag{1.10}\end{equation*}\)
Since we have two constants to determine, more information is needed. Differentiating (1.9), we have
\(\frac{d x}{d t}=v=\frac{d}{d t} g\left(t, c_{1}, c_{2}\right)\)
If we also know that at time $t_{0}$ the particle has velocity $v_{0}$, then we have the additional relation
\(\begin{equation*}v_{0}=\left.\frac{d}{d t} g\left(t, c_{1}, c_{2}\right)\right|_{t=t_{0}} \tag{1.11}\end{equation*}\)
We may then use (1.10) and (1.11) to solve for $c_{1}$ and $c_{2}$ in terms of $x_{0}$ and $v_{0}$. Knowing $c_{1}$ and $c_{2}$, we can use Eq. (1.9) to predict the exact future motion of the particle.
As an example of Eqs. (1.8) to (1.11), consider the vertical motion of a particle in the earth's gravitational field. Let the $x$ axis point upward. The force on the particle is downward and is $F=-m g$, where $g$ is the gravitational acceleration constant. Newton's second law (1.8) is $-m g=m d^{2} x / d t^{2}$, so $d^{2} x / d t^{2}=-g$. A single integration gives $d x / d t=-g t+c_{1}$. The arbitrary constant $c_{1}$ can be found if we know that at time $t_{0}$ the particle had velocity $v_{0}$. Since $v=d x / d t$, we have $v_{0}=-g t_{0}+c_{1}$ and $c_{1}=v_{0}+g t_{0}$. Therefore, $d x / d t=-g t+g t_{0}+v_{0}$. Integrating a second time, we introduce another arbitrary constant $c_{2}$, which can be evaluated if we know that at time $t_{0}$ the particle had position $x_{0}$. We find (Prob. 1.7) $x=x_{0}-\frac{1}{2} g\left(t-t_{0}\right)^{2}+v_{0}\left(t-t_{0}\right)$. Knowing $x_{0}$ and $v_{0}$ at time $t_{0}$, we can predict the future position of the particle.
The classical-mechanical potential energy $V$ of a particle moving in one dimension is defined to satisfy
\(\begin{equation*}\frac{\partial V(x, t)}{\partial x}=-F(x, t) \tag{1.12}\end{equation*}\)
For example, for a particle moving in the earth's gravitational field, $\partial V / \partial x=-F=m g$ and integration gives $V=m g x+c$, where $c$ is an arbitrary constant. We are free to set the zero level of potential energy wherever we please. Choosing $c=0$, we have $V=m g x$ as the potential-energy function.
The word state in classical mechanics means a specification of the position and velocity of each particle of the system at some instant of time, plus specification of the forces
acting on the particles. According to Newton's second law, given the state of a system at any time, its future state and future motions are exactly determined, as shown by Eqs. (1.9)-(1.11). The impressive success of Newton's laws in explaining planetary motions led many philosophers to use Newton's laws as an argument for philosophical determinism. The mathematician and astronomer Laplace (1749-1827) assumed that the universe consisted of nothing but particles that obeyed Newton's laws. Therefore, given the state of the universe at some instant, the future motion of everything in the universe was completely determined. A super-being able to know the state of the universe at any instant could, in principle, calculate all future motions.
#### Abstract
Although classical mechanics is deterministic, many classical-mechanical systems (for example, a pendulum oscillating under the influence of gravity, friction, and a periodically varying driving force) show chaotic behavior for certain ranges of the systems' parameters. In a chaotic system, the motion is extraordinarily sensitive to the initial values of the particles' positions and velocities and to the forces acting, and two initial states that differ by an experimentally undetectable amount will eventually lead to very different future behavior of the system. Thus, because the accuracy with which one can measure the initial state is limited, prediction of the long-term behavior of a chaotic classical-mechanical system is, in practice, impossible, even though the system obeys deterministic equations. Computer calculations of solar-system planetary orbits over tens of millions of years indicate that the motions of the planets are chaotic [I. Peterson, Newton's Clock: Chaos in the Solar System, Freeman, 1993; J. J. Lissauer, Rev. Mod. Phys., 71, 835 (1999)].
Given exact knowledge of the present state of a classical-mechanical system, we can predict its future state. However, the Heisenberg uncertainty principle shows that we cannot determine simultaneously the exact position and velocity of a microscopic particle, so the very knowledge required by classical mechanics for predicting the future motions of a system cannot be obtained. We must be content in quantum mechanics with something less than complete prediction of the exact future motion.
Our approach to quantum mechanics will be to postulate the basic principles and then use these postulates to deduce experimentally testable consequences such as the energy levels of atoms. To describe the state of a system in quantum mechanics, we postulate the existence of a function $\Psi$ of the particles' coordinates called the state function or wave function (often written as wavefunction). Since the state will, in general, change with time, $\Psi$ is also a function of time. For a one-particle, one-dimensional system, we have $\Psi=\Psi(x, t)$. The wave function contains all possible information about a system, so instead of speaking of "the state described by the wave function $\Psi$," we simply say "the state $\Psi$." Newton's second law tells us how to find the future state of a classicalmechanical system from knowledge of its present state. To find the future state of a quantum-mechanical system from knowledge of its present state, we want an equation that tells us how the wave function changes with time. For a one-particle, one-dimensional system, this equation is postulated to be
\(\begin{equation*}-\frac{\hbar}{i} \frac{\partial \Psi(x, t)}{\partial t}=-\frac{\hbar^{2}}{2 m} \frac{\partial^{2} \Psi(x, t)}{\partial x^{2}}+V(x, t) \Psi(x, t) \tag{1.13}\end{equation*}\)
where the constant $\hbar$ (h-bar) is defined as
\(\begin{equation*}\hbar \equiv \frac{h}{2 \pi} \tag{1.14}\end{equation*}\)
The concept of the wave function and the equation governing its change with time were discovered in 1926 by the Austrian physicist Erwin Schrödinger (1887-1961). In this equation, known as the time-dependent Schrödinger equation (or the Schrödinger wave equation), $i=\sqrt{-1}, m$ is the mass of the particle, and $V(x, t)$ is the potentialenergy function of the system. (Many of the historically important papers in quantum mechanics are available at dieumsnh.qfb.umich.mx/archivoshistoricosmq.)
The time-dependent Schrödinger equation contains the first derivative of the wave function with respect to time and allows us to calculate the future wave function (state) at any time, if we know the wave function at time $t_{0}$.
The wave function contains all the information we can possibly know about the system it describes. What information does $\Psi$ give us about the result of a measurement of the $x$ coordinate of the particle? We cannot expect $\Psi$ to involve the definite specification of position that the state of a classical-mechanical system does. The correct answer to this question was provided by Max Born shortly after Schrödinger discovered the Schrödinger equation. Born postulated that for a one-particle, one-dimensional system,
\(\begin{equation*}|\Psi(x, t)|^{2} d x \tag{1.15}\end{equation*}\)
gives the probability at time $t$ of finding the particle in the region of the $x$ axis lying between $x$ and $x+d x$. In (1.15) the bars denote the absolute value and $d x$ is an infinitesimal length on the $x$ axis. The function $|\Psi(x, t)|^{2}$ is the probability density for finding the particle at various places on the $x$ axis. (Probability is reviewed in Section 1.6.) For example, suppose that at some particular time $t_{0}$ the particle is in a state characterized by the wave function $a e^{-b x^{2}}$, where $a$ and $b$ are real constants. If we measure the particle's position at time $t_{0}$, we might get any value of $x$, because the probability density $a^{2} e^{-2 b x^{2}}$ is nonzero everywhere. Values of $x$ in the region around $x=0$ are more likely to be found than other values, since $|\Psi|^{2}$ is a maximum at the origin in this case.
To relate $|\Psi|^{2}$ to experimental measurements, we would take many identical noninteracting systems, each of which was in the same state $\Psi$. Then the particle's position in each system is measured. If we had $n$ systems and made $n$ measurements, and if $d n_{x}$ denotes the number of measurements for which we found the particle between $x$ and $x+d x$, then $d n_{x} / n$ is the probability for finding the particle between $x$ and $x+d x$. Thus
\(\frac{d n_{x}}{n}=|\Psi|^{2} d x\)
and a graph of $(1 / n) d n_{x} / d x$ versus $x$ gives the probability density $|\Psi|^{2}$ as a function of $x$. It might be thought that we could find the probability-density function by taking one system that was in the state $\Psi$ and repeatedly measuring the particle's position. This procedure is wrong because the process of measurement generally changes the state of a system. We saw an example of this in the discussion of the uncertainty principle (Section 1.3).
Quantum mechanics is statistical in nature. Knowing the state, we cannot predict the result of a position measurement with certainty; we can only predict the probabilities of various possible results. The Bohr theory of the hydrogen atom specified the precise path of the electron and is therefore not a correct quantum-mechanical picture.
Quantum mechanics does not say that an electron is distributed over a large region of space as a wave is distributed. Rather, it is the probability patterns (wave functions) used to describe the electron's motion that behave like waves and satisfy a wave equation.
How the wave function gives us information on other properties besides the position is discussed in later chapters.
The postulates of thermodynamics (the first, second, and third laws of thermodynamics) are stated in terms of macroscopic experience and hence are fairly readily understood. The postulates of quantum mechanics are stated in terms of the microscopic world and appear quite abstract. You should not expect to fully understand the postulates of quantum mechanics at first reading. As we treat various examples, understanding of the postulates will increase.
It may bother the reader that we wrote down the Schrödinger equation without any attempt to prove its plausibility. By using analogies between geometrical optics and classical mechanics on the one hand, and wave optics and quantum mechanics on the other hand, one can show the plausibility of the Schrödinger equation. Geometrical optics is an approximation to wave optics, valid when the wavelength of the light is much less than the size of the apparatus. (Recall its use in treating lenses and mirrors.) Likewise, classical mechanics is an approximation to wave mechanics, valid when the particle's wavelength is much less than the size of the apparatus. One can make a plausible guess as to how to get the proper equation for quantum mechanics from classical mechanics based on the known relation between the equations of geometrical and wave optics. Since many chemists are not particularly familiar with optics, these arguments have been omitted. In any case, such analogies can only make the Schrödinger equation seem plausible. They cannot be used to derive or prove this equation. The Schrödinger equation is a postulate of the theory, to be tested by agreement of its predictions with experiment. (Details of the reasoning that led Schrödinger to his equation are given in Jammer, Section 5.3. A reference with the author's name italicized is listed in the Bibliography.)
Quantum mechanics provides the law of motion for microscopic particles. Experimentally, macroscopic objects obey classical mechanics. Hence for quantum mechanics to be a valid theory, it should reduce to classical mechanics as we make the transition from microscopic to macroscopic particles. Quantum effects are associated with the de Broglie wavelength $\lambda=h / m v$. Since $h$ is very small, the de Broglie wavelength of macroscopic objects is essentially zero. Thus, in the limit $\lambda \rightarrow 0$, we expect the time-dependent Schrödinger equation to reduce to Newton's second law. We can prove this to be so (see Prob. 7.59).
A similar situation holds in the relation between special relativity and classical mechanics. In the limit $v / c \rightarrow 0$, where $c$ is the speed of light, special relativity reduces to classical mechanics. The form of quantum mechanics that we will develop will be nonrelativistic. A complete integration of relativity with quantum mechanics has not been achieved.
Historically, quantum mechanics was first formulated in 1925 by Heisenberg, Born, and Jordan using matrices, several months before Schrödinger's 1926 formulation using differential equations. Schrödinger proved that the Heisenberg formulation (called matrix mechanics) is equivalent to the Schrödinger formulation (called wave mechanics). In 1926, Dirac and Jordan, working independently, formulated quantum mechanics in an abstract version called transformation theory that is a generalization of matrix mechanics and wave mechanics. In 1948, Feynman devised the path integral formulation of quantum mechanics [R. P. Feynman, Rev. Mod. Phys., 20, 367 (1948); R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, 1965].