Question

In Section \(5.3\), we leamed that to be normaliable, a wave function (1) must not itself diverge and \((2)\) must fall to 0 faster than \(|\mathrm{r}|^{-1 / 2}\) as \(x\) gets lar ge. Nevertheless, we find two functions that slightly violate these requirements very useful. Consider the quantum mechanical plane wave Aelue-aul and the weird function \(\psi_{x_{1}}(x)\) pictured in Figure \(5.19\). which we here call by its preper name, the Direc delta function. (a) Which of the iwo normalizability requirements is violated by the plane wave, and which by the Dirac delta function? (b) Nomalization of the plane Wave could be accomplished if it were simply truncated, restricted to the rejion \(-b

Short answer

The plane wave violates the condition of falling to 0 faster than \(|x|^{-1/2}\) as x gets large, whereas the Dirac delta function diverges at x=0. The normalization relationship for the truncated plane wave is \(A = \sqrt{1/(2b)}\) and for the new box function is \(B = \sqrt{1/(2ε)}\). The plane wave has certain momentum but uncertain position. The Dirac delta function has certain position but uncertain momentum, hence they are complementary in terms of position and momentum uncertainties.

Step-by-step solution

Identifying Normalizability Violations

For a plane wave, it violates the normalizability condition that it should fall to 0 faster than \(|x|^{-1/2}\) as x gets large. It doesn't fall to 0 at all, remaining constant instead, hence the condition is violated. For the Dirac delta function, it violates the condition that the function itself should not diverge; the Dirac delta function diverges at x=0.

Normalizing Plane Wave

To normalize the plane wave, it could be truncated at -b and +b, yielding a box function, which can be normalized since it's a finite function bounded to the region (-b,b). The relation between A and b will be \(A = \sqrt{1/(2b)}\) since for normalization, the integral of |Ae^(ikx)|^2 from -b to b should equal 1. Now, as b approaches infinity, A will approach 0.

Normalizing the New Function

Consider the function is 0 everywhere except for (-ε,ε) where its value is a constant B, this function too can be normalized. The relationship between ε and B would be \(B = \sqrt{1/(2ε)}\). Now, as ε approaches 0, B will approach infinity.

Complementarity of Functions

The plane wave and Dirac delta functions are complementary to each other in terms of Heisenberg's Uncertainty principles. For the plane wave, momentum is certain (zero uncertainty), but position is completely uncertain. Conversely, for the Dirac delta function, position is certain and momentum is completely uncertain.

Key concepts

Understanding Quantum Mechanics

Quantum mechanics is a fundamental theory that describes the behavior of particles at atomic and subatomic levels. Unlike classical physics, quantum mechanics reveals that particles can exhibit wave-like properties, leading to concepts such as wave-particle duality. Crucial to the theory is the wave function, which mathematically represents the quantum state of a particle or system of particles. A key aspect of a valid wave function is normalizability, meaning it must be mathematically integrable to one over all space, representing a total probability of finding the particle somewhere in space.
When assessing whether a wave function is normalizable, it must adhere to two conditions: it should not diverge, and it must decay faster than the rate of \(|\text{r}|^{-1/2}\) as the position \(x\) becomes large. These conditions ensure that the probability of finding a particle doesn't approach infinity and is confined to a finite region of space, aligning with the probabilistic interpretation of quantum mechanics.

Plane Waves in Quantum Mechanics

In quantum mechanics, a plane wave represents the wave function of a free particle with a definite momentum, described by the expression \(Ae^{i(kx - \text{ωt})}\). This expression characterizes a wave that extends infinitely in space without change in amplitude, making it non-normalizable by the standard criteria. This is because it does not diminish as \(x\) increases; it remains constant, violating one of the requirements for normalizability. However, to make the plane wave normalizable, it can be truncated within a finite region, creating a box function, effectively constraining the wave to a limited region of space to allow for normalization. Truncating the wave imposes boundary conditions that discretize its momentum values, creating a more physically realistic scenario where the particle's location is limited to a certain region yet remains uncertain within that region, adhering to the principles of quantum mechanics.

Dirac Delta Function and its Role

The Dirac delta function is a unique and powerful tool in quantum mechanics and mathematical physics, commonly used to represent an infinitely sharp spike at a single point in space, effectively describing a particle located precisely at that point. Mathematically, it is denoted as \(\text{δ}(x-x_0)\), where it equals zero for all values of \(x\) except at \(x = x_0\), where it diverges. This function is not a function in the traditional sense, but rather a 'distribution' or 'generalized function' used within integrals.
While the Dirac delta function itself is not normalizable because it does not satisfy the criterion of the wave function not diverging, it can be approached as the limit of a normalizable function. For example, when considering a box function of height B and width \(2ε\) that is non-zero only within a narrow region around the point, as \(ε\) approaches zero, the function begins to resemble a Dirac delta function. Conceptually, it serves to model situations in quantum mechanics where extreme localization is necessary, such as representing the precise position of a particle.

Heisenberg's Uncertainty Principle Explained

Heisenberg's Uncertainty Principle is a cornerstone of quantum mechanics articulating a fundamental limit to the precision with which pairs of physical properties, like position (\(x\)) and momentum (\(p\)), can be known simultaneously. The principle asserts that the more precisely the position of a particle is determined, the less precisely its momentum can be known, and vice versa. Mathematically, this is expressed as \(ΔxΔp ≥ \frac{ℏ}{2}\), where \(Δx\) and \(Δp\) are the uncertainties in position and momentum, respectively, and \(ℏ\) is the reduced Planck’s constant.
This principle has profound implications on the nature of wave functions, particularly in the context of the plane wave and the Dirac delta function. The plane wave, with a precisely defined momentum, implies an infinite uncertainty in position – a particle could be anywhere. Conversely, the Dirac delta function, which precisely pinpoints a particle's position, implies infinite uncertainty in momentum – the particle's momentum can have any value. Therefore, these two functions exemplify the core tenets of Heisenberg's Uncertainty Principle, illustrating the unavoidable trade-off between the certainty of a particle's position and momentum in the quantum realm.