10.2: Introduction to Quantum Mechanics

Perhaps the first question to ask about a physical object is, “where is it?” In everyday experience, it is possible to answer that question with great precision, limited only by the quality of measurement apparatus. In the realm of very small objects, however, there are some fundamental limitations and quantum mechanics must be used to address that question.

At its heart, quantum mechanics deals with energy. Because of the equivalence of mass and energy (remember Einstein’s famous formula \(E = mc^2\) where \(c\) is the speed of light, \(2.998 × 10^8\) meters per second) quantum mechanics also deals with particles with mass. And because of the relationship between energy of a photon and its frequency (\(E = hf\) where \(h\) is the Planck constant, \(6.626 × 10^{−34}\) Joule-seconds) quantum mechanics deals with photons.

According to quantum mechanics, the question “where is it” cannot be answered with certainty. How do we deal with uncertainty? By assigning probabilities. It is a little more complicated because of the continuous nature of space, and because space is considered to be infinite in extent (at least if general relativity is ignored), but the idea is the same as for probabilities of a finite set of events. The probability density is nonnegative, and integrates over all space to 1 (this is like the sum of the probabilities of all events that are mutually exclusive and exhaustive adding up to 1).

Thus in quantum mechanics, an object is represented as a “probability blob” which evolves over time. How does it evolve? The underlying equation is not written in terms of the probability density, but rather in terms of another function of space and time from which the probability density can be found.

Consider the square root of the probability density, as a function of space and time. Then, for added generality, let the square root be either positive or negative—when you square it to get the probability density, either one will do. Next, for even more generality, allow this square root to have an arbitrary phase in the complex plane, so that it has both a real and an imaginary part. We will no longer call this the square root, but instead the “wave function” \(\psi(r, t)\) which is a complex-valued function of space \(r\) and time \(t\). The probability density is then the magnitude of the wave function squared

\(|\psi(r, t)|^2 = \psi(r, t)\psi ^* (r, t) \tag{10.1}\)

where the asterisk * denotes the complex conjugate.

In dealing with probabilities earlier, we never expressed them in terms of anything more primitive. Why do we need to now? Because the fundamental equation of quantum mechanics deals with \(\psi (r, t)\) rather than the probability density. Why is this? Don’t ask. It’s just one of many bizarre features of quantum mechanics.

The fundamental equation of quantum mechanics is the Schrödinger equation, published in 1926 by the Austrian physicist Erwin Schrödinger (1887–1961).\(^1\)

where \(i\) is the (imaginary) square root of -1, \(m\) is the mass of this object, \(V(r)\) is the potential energy function, and \(\hbar = h/2\pi = 1.054 × 10^{−34}\) Joule-seconds. Note that this equation contains partial derivatives in both space and time. The derivative with respect to time is first order, and the spatial derivatives are second order. The Laplacian \(\nabla ^2\) is defined as

\(\nabla ^2f = \dfrac {\partial ^2 f}{\partial x^2} + \dfrac {\partial ^2 f}{\partial y^2} + \dfrac {\partial ^2 f}{\partial z^2} \tag{10.3}\)

where \(x\), \(y\), and \(z\) are the three spacial dimensions.

This equation 10.2 is frequently interpreted by multiplying both sides of it by \(\psi ^*(r, t)\) and integrating over space. Then the left-hand side is identified as the total energy, and the right-hand side as the sum of the kinetic and potential energies (assuming the wave function is normalized so that the space integral of \(|\psi (r, t)|^2\) is 1, a property required for the interpretation in terms of a probability density). Along with this interpretation it is convenient to call \(i\hbar \partial/\partial t\) the energy operator. It is an operator in the mathematical sense (something that operates on a function and produces as a result a function) and it has the correct dimensional units to be energy. Quantum mechanics is often formulated in terms of similar operators.

The Schrödinger equation is deceptively simple. It is a linear equation in \(\psi (r, t)\) in the sense that if both \(\psi_1\) and \(\psi_2\) are solutions then so is any linear combination of them

\(\psi_{\text{total}} = \alpha_1 \psi_1 + \alpha_2 \psi_2 \tag{10.4}\)

where \(\alpha_1\) and \(\alpha_2\) are complex constants (if the linear combination is to lead to a valid probability distribution then the values of \(\alpha_1\) and \(\alpha_2\) must be such that the integral over all space of \(|\psi_{\text{total}}|^2\) is 1). However, except for the simplest cases of \(V(r)\) the equation has not been solved in closed form.

Strictly speaking, the Schrödinger equation is really only correct if the object being described is the entire universe and \(V(r)\) = 0, in which case the equation is useless because it is so complicated. However, it is often used as an approximation in the case where the universe is considered in two pieces—a small one (the object) whose wave function is being calculated, and the rest of the universe (the “environment”) whose influence on the object is assumed to be represented by the \(V (r)\) term. Note that the object may be a single photon, a single electron, or two or more particles, i.e., it need not correspond to the everyday concept of a single particle.

An object might interact with its environment. Naturally, if the object changes its environment (as would happen if a measurement were made of some property of the object) then the environment in turn would change the object. Thus after a measurement, an object will generally have a different wave function, and some information about the object may no longer be accessible. It is a feature of quantum mechanics that this new wave function is consistent with the changes in the environment; whether this feature is a consequence of the Schrödinger equation or is a separate aspect of quantum mechanics is unclear.

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\(^1\)See a biography of Schrödinger at mathshistory.st-andrews.ac.uk/Biographies/Schrödinger/