Schrödinger's
Wave Function
A Journey Into Quantum World
Overview of the
Concept, Implications, and Paradox
Schrödinger’s wave function is a fundamental concept in quantum mechanics, encapsulating the probabilities of a particle’s properties, such as position and momentum. This concept has profound implications, leading to phenomena such as quantum superposition and quantum entanglement, and culminating in the famous thought experiment known as Schrödinger’s cat paradox.
Quantum Physics, also known as Quantum Mechanics, refers to the branch of physics that deals with the behavior of things at the subatomic level, the world that we are not very familiar with in our day today lives. The laws of physics at the microscopic level, unlike at the macroscopic levels that we are accustomed to, are very strange and counter-intuitive.
In 1925 an Austrian physicist Erwin Schrödinger, in an attempt to describe a quantum system like an electron in an atom, revolutionized the field of quantum physics with his formulation of the wave function.
Let us explore this weird quantum world and how the discovery of wave function, by Schrodinger, led to profound and strange implications.
De Broglie’s Matter Waves
Albert Einstein, in his explanation of the photoelectric effect, discovered that light possesses both wave-like and particle-like properties. Conversely, as discussed in our post on wave-particle duality, de Broglie proposed that particles such as electrons could exhibit wave-like characteristics.
By applying Einstein’s photoelectric equation and the special theory of relativity, de Broglie theoretically derived an important equation. He demonstrated that all particles, including electrons, have associated wave properties, and their wavelengths are described by this equation.
In physics, every wave—from sound to electromagnetic—has an equation. However, de Broglie’s matter-waves lacked a wave equation, which he did not attempt to derive.
Wave Function
Origin and Formulation
In 1925, Erwin Schrödinger took up this task of deriving the wave equation for a particle such an electron in a hydrogen atom.
In 1926 Schrödinger published 256 pages in which he arrived at his wave equation through a series of thought experiments and mathematical derivations, building on the work of earlier physicists.
Max Planck’s quantization of energy levels and Albert Einstein’s work on the photoelectric effect provided crucial insights into the dual particle-wave nature of matter. However, it was Louis de Broglie’s hypothesis that particles like electrons could exhibit wave-like properties that directly influenced Schrödinger’s work.
Inspired by de Broglie’s wave-particle duality, Schrödinger postulated that a wave function, typically denoted as Ψ (Psi), described the behavior of quantum particles. He formulated the time-dependent Schrödinger equation, which is a partial differential equation that describes how the wave function evolves over time:
Where, “I” is the imaginary unit, “ħ” (h-bar) is the reduced Planck’s constant, and the right-hand side represents the total energy of the system.
Schrödinger’s
Wave Equation
Schrödinger developed the wave equation based on de Broglie’s wave-particle duality, which connects a particle’s wavelength to its momentum, along with established classical physics equations. This wave equation formed the basis of wave mechanics. He needed to demonstrate that it was indeed the correct wave equation. By applying it to the hydrogen atom, he aimed to see if it produced accurate values for the energy levels.
Schrödinger subsequently discovered that his wave equation yielded a series of energy levels for the hydrogen atom. Unlike de Broglie’s one-dimensional standing electron waves, which were fitted into circular orbits, Schrödinger’s theory derived their three-dimensional counterparts—electron orbitals. The corresponding energies were an inherent part of the acceptable solutions to Schrödinger’s wave equation.
This development rendered obsolete the ad hoc modifications required by the Bohr-Sommerfield quantum model; the previous adjustments were now naturally integrated within the framework of Schrödinger’s wave mechanics. Additionally, the smooth, continuous transitions between permitted three-dimensional electron standing waves replaced the enigmatic quantum jumps between electron orbitals.
Implications of
Schrödinger’s Wave-Function
When Schrödinger discovered the wave equation for de Broglie matter waves, the wave function was the unknown part. Solving the equation for a particular physical situation such as the hydrogen atom, would yield the wave function. However, there was a question that Schrödinger was finding difficult to answer: what was doing the waving?
If Schrödinger’s wave function did not represent real waves in everyday 3-dimensional space, then what were they?
The German British physicist Max Born provided the answer.
According to Schrödinger, there were no quantum jumps between different energy levels in an atom; instead, there was a smooth, continuous transition from one standing wave to another, with the emission of radiation resulting from a resonance phenomenon. He proposed that wave mechanics permitted a depiction of physical reality characterized by continuity, causality, and determinism.
But Max Born disagreed. He accepted the richness of Schrödinger’s formalism but rejected his interpretation.
Born utilized Schrödinger’s framework to present a fundamentally different concept, revealing that probability is central to wave mechanics and quantum theory. He put forward an interpretation of the wave function that challenged a fundamental tenet of physics – determinism.
He applied wave mechanics to depict a picture of reality characterized by discontinuity, acausality, and probability. These two representations of reality depend on different interpretations of the wave function, symbolized by the Greek letter psi (Ψ) in Schrödinger’s wave equation.
Born stated that the square of the wave function pertains to the realm of probability. Squaring the wave function, for instance, does not yield the actual position of an electron but the probability that it will be found here rather than there.
According to Born, Schrödinger’s equation represents a probability wave. This implies that there are no actual electron waves, but rather abstract waves of probability.
Copenhagen Interpretation
Danish physicist Niels Bohr would soon argue that until an observation or measurement is made, a microphysical object like an electron does not have a definite location. Between one measurement and the next, it exists only within the abstract possibilities described by the wave function.
When an observation or measurement occurs, the wave function collapses as one of the possible states of the electron becomes the actual state, while the probability of all other possibilities becomes zero.
Einstein’s Concern
Soon after Bohr’s interpretation of the wave function, Einstein expressed his increasing concern regarding the dismissal of causality and determinism.
He stated that, which went like this:
“Quantum mechanics is currently formidable. However, an inner conviction tells me that it has not yet revealed the ultimate truth. The theory provides extensive information but fails to truly advance our understanding of the fundamental nature of the universe. I remain convinced that ‘God does not play dice.'”
Quantum Superposition
A significant implication of Schrödinger’s wave function is the principle of quantum superposition. This principle posits that a quantum system can concurrently exist in multiple states until it is observed or measured. The wave function Ψ encompasses all potential states of the system, and the process of measurement results in the collapse of the wave function to a single state.
For instance, an electron in an atom exists within a cloud of probabilities characterized by its wave function, rather than occupying a single orbit. The measurement of its location determines the electron’s specific location.
Quantum Entanglement
Another profound implication of Schrödinger’s wave function is quantum entanglement. When two particles become entangled, their properties become interconnected, such that the state of one particle instantly influences the state of the other, regardless of the distance between them. Einstein famously described this phenomenon as “spooky action at a distance.”
Entanglement arises naturally from the solutions to Schrödinger’s wave equation for systems of multiple particles. Experimental confirmation has established it as the foundation for developing technologies such as quantum computing and quantum cryptography.
Schrödinger’s Cat Paradox
Despite its successes, the interpretation of the wave function and its implications resulted in philosophical discussions, illustrated by Schrödinger’s cat paradox.
Schrödinger proposed a thought experiment in which a cat is placed in a sealed box with a radioactive atom, a Geiger counter, a vial of poison, and a hammer. If the Geiger counter detects radiation (i.e., the atom decays), the hammer gets released, breaking the vial and killing the cat. If it does not detect radiation, the cat remains alive.
According to quantum mechanics, until the box is opened, and an observation is made, the radioactive atom is in a superposition of decayed and undecayed states. Consequently, the cat is simultaneously alive and dead, a paradox that challenges our classical understanding of reality.
Conclusion
Schrödinger’s wave function has reshaped our understanding of the quantum world, unveiling phenomena like superposition and entanglement that defy classical intuition.
The implications of this theory extend beyond physics, influencing fields such as information technology and philosophy.
Schrödinger’s cat paradox continues to provoke thought and debate, highlighting the unresolved questions at the heart of quantum mechanics. As research advances, the wave function remains a central element in the quest to comprehend the mysteries of the quantum realm.
References
- This Biggest Ideas in the Universe – Quanta and Fields: By Sean Carroll
- Quantum by Manjit Kumar
- Einstein by Walter Isaacson.
- https://youtu.be/2WPA1L9uJqo