Question

In the 1920 s, de Broglie and Schröinger developed what is now called wave mechanics or quantum mechanics. Which of the following statements with regard to this model is(are) true? a. The position of an electron can be exactly found and measured. b. With the use of modern instruments, scientists can watch an electron move from point to point in its orbit. c. Energy levels in quantum mechanics are not limited to certain discrete values but are actually continuous. A. An electron has properties of a particle and a wave in this wave mechanical model of the atom.

Short answer

None of the given statements (a, b, and c) are true regarding wave mechanics or quantum mechanics. Statement a is false due to Heisenberg's uncertainty principle, statement b is also false because of the uncertainty principle, and statement c is false as energy levels in quantum mechanics are quantized and not continuous.

Step-by-step solution

Analyze Statement a

The first statement states that in the wave mechanics model, the position of an electron can be exactly found and measured. However, Heisenberg's uncertainty principle contradicts this statement. According to the uncertainty principle, we cannot know the exact position and momentum of an electron simultaneously. Thus, this statement is false.

Analyze Statement b

Statement b says that with the use of modern instruments, scientists can watch an electron move from point to point in its orbit. This statement is also false, as observing the electron's movement would require knowing its exact position and momentum simultaneously, which is not possible due to Heisenberg's uncertainty principle.

Analyze Statement c

The third statement asserts that energy levels in quantum mechanics are not limited to certain discrete values but are actually continuous. This statement is false, as one of the primary aspects of quantum mechanics is the quantized nature of energy levels. This means that energy levels are restricted to discrete values and are not continuous.

Determine the correct answer

As we have analyzed all three statements and determined that all of them are false, the correct answer to this question is that none of the given statements are true regarding wave mechanics or quantum mechanics.

Key concepts

Wave Mechanics

Wave mechanics, developed in the early 20th century, revolutionized our understanding of atomic and subatomic systems. It forms the basis of quantum mechanics—a theory that interprets particles like electrons as having wave-like properties.

Consider the behavior of electrons in an atom. Traditional models depicted electrons traveling in fixed orbits, akin to planets orbiting a sun. However, wave mechanics suggests a fundamentally different picture. Electrons are described by wave functions, mathematical representations that give the probabilities of finding an electron in particular locations. Unlike the certainty of planetary orbits, electron positions are inherently uncertain—they exist in a 'probability cloud' around the nucleus.

To visualize this, imagine throwing a pebble into a pond. The ripples spreading out represent the electron's wave function, hinting at where the electron might be found, but not exactly pinpointing a location. The undulatory nature of these particles challenges the classical understanding and is one of the cornerstones of quantum mechanics.

Heisenberg's Uncertainty Principle

One of the most iconic and fundamental concepts in quantum mechanics is Heisenberg's uncertainty principle. This principle places a limit on our ability to know both the position and the momentum of a particle, such as an electron, at the same time.

Heisenberg's principle can be mathematically expressed as \( \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \), where \( \Delta x \) is the uncertainty in position, \( \Delta p \) is the uncertainty in momentum, and \( \hbar \) is the reduced Planck constant. The principle suggests that the more precisely we know the position of a quantum particle, the less precisely we can know its momentum, and vice versa.

This runs counter to classical mechanics where, theoretically, both can be measured to infinite precision. The uncertainty principle is not a limitation of our instruments; rather, it's a fundamental aspect of the natural world at the quantum scale. It tells us that the act of measuring an electron's position changes its momentum, making the simultaneous knowledge of both impossibilities.

Quantized Energy Levels

In quantum mechanics, energy levels of atoms and molecules are quantized, meaning that they are restricted to certain discrete values and are not continuous. This phenomenon was one of the radical changes quantum theory brought to physics, challenging the formerly held belief that energies could vary smoothly.

The concept of quantization was first introduced by Max Planck and later developed by others like Albert Einstein and Niels Bohr. It's exemplified in the simplest atom, hydrogen, where an electron can only occupy certain energy levels. When an electron transitions between these levels, photons are absorbed or emitted, corresponding to the difference in energy levels.

The quantization of energy levels is not arbitrary; it results from the wave-like properties of particles. Similar to how a string fixed at both ends can only support standing waves at certain frequencies, electrons in atoms can only exist at certain energy levels, representing 'standing waves' of probability.

Electron Particle-Wave Duality

Electron particle-wave duality is a central tenet of quantum mechanics, encapsulating the theory that particles can exhibit both particle and wave characteristics. This complex concept is often illustrated via the famous double-slit experiment, where electrons show interference patterns indicative of waves when not observed, but when one attempts to measure which slit an electron passes through, it behaves like a particle.

This duality means that while we can talk about the position of an electron (particle nature), its full description requires understanding its wave function (wave nature). It also underscores a fundamental aspect of quantum systems: observation affects the observed, as measuring certain properties of quantum objects can alter their behaviour.

In sum, while at macroscopic scales objects appear to have clearly defined locations and trajectories, at the quantum level, entities like electrons are better conceived as existing in a superposition of states until a measurement is made, transforming potential into actuality. This duality is not a simple paradox but is instead evidence of the richness and complexity inherent in the fabric of the physical world.