Davisson-Germer Experiment: An Experiment that confirms the existence of de Broglie waves.

 The Davisson-Germer Experiment is a key experiment that confirms the wave nature of particles, specifically electrons, as predicted by de Broglie. This experiment demonstrates that particles like electrons can exhibit diffraction, a property of waves, which supports the existence of de Broglie waves. 

What is de Broglie’s Hypothesis?

In 1924, Louis de Broglie proposed that all matter has wave-like properties. He suggested that the wavelength (λ) of a particle is related to its momentum (p) by the formula: 

Where:

•  = wavelength of the particle
•  = Planck’s constant ()
•  = momentum of the particle (, where  is mass and  is velocity)

This idea led to the concept of matter waves (also called de Broglie waves).

Davisson-Germer Experiment Overview

The Davisson-Germer experiment was conducted in 1927 by Clinton Davisson and Lester Germer. It aimed to study how electrons scatter off a crystal surface. The unexpected result was the discovery of electron diffraction, proving that electrons have wave-like behavior, just as light does.

Setup of the Experiment

• Electron gun: This emits a beam of electrons.
• Nickel target: A nickel crystal acts as a diffraction grating.
• Electron detector: Measures the intensity of scattered electrons at different angles.
• Accelerating voltage: Controls the speed (and thus the momentum) of the electrons.

How the Experiment Works

1. Electron emission: Electrons are emitted from an electron gun and accelerated by a potential difference (V). The kinetic energy of the electrons is given by:

   
   

   Where:

   • ​ is the kinetic energy of the electrons
   •  is the charge of the electron ()
   •  is the accelerating voltage

2. Momentum of electrons: The momentum of an electron is related to its kinetic energy:

   ​

   Where:

   •  is the mass of the electron ()
   •  is the accelerating voltage

3. Electron diffraction: When the electron beam strikes the nickel crystal, the atoms of the crystal scatter the electrons. The crystal structure acts like a diffraction grating for the electron waves.

4. Measurement of angles: The scattered electrons are detected at various angles, and the intensity of the scattered electrons is measured. A sharp peak in intensity occurs at specific angles, showing constructive interference, a key sign of wave behavior.

Bragg’s Law

The observed diffraction pattern can be explained by Bragg’s law, which relates the angle of diffraction (θ) to the wavelength of the electrons and the spacing between the crystal planes (d):

Where:

•  = order of the diffraction (usually  for the first-order diffraction)
• d = spacing between crystal planes
•  = angle of incidence that results in constructive interference
•  = wavelength of the electron (from de Broglie’s equation)

Verifying de Broglie’s Hypothesis

Using the de Broglie wavelength for the electrons:

​

By adjusting the accelerating voltage (V), the wavelength of the electrons can be changed. The diffraction pattern observed at different angles confirms that the electrons behave like waves, with their wavelength matching de Broglie’s prediction.

Results of the Experiment

At a specific accelerating voltage (around 54V), a sharp diffraction peak was observed at an angle of about 50°. Using Bragg’s law, the electron wavelength was calculated and found to match the de Broglie wavelength, confirming the wave nature of electrons. 

Key Takeaways for Students:

1. Wave-particle duality: The Davisson-Germer experiment confirms that particles such as electrons can behave as waves, supporting de Broglie’s hypothesis.
2. Diffraction pattern: The diffraction of electrons off the nickel crystal proves that particles can undergo constructive and destructive interference, a wave-like property.
3. De Broglie wavelength: The experiment provides experimental evidence for the de Broglie wavelength of matter waves.

This experiment is crucial because it supports quantum mechanics' view that matter, on a small scale, behaves as both particles and waves. 

Schrödinger’s Cat Experiment: A Quantum Mystery.

Schrödinger’s Cat Experiment: A Quantum Mystery. 

The Schrödinger's Cat experiment is one of the most famous thought experiments in the field of quantum mechanics, proposed by Austrian physicist Erwin Schrödinger in 1935. The experiment was designed to illustrate the peculiarities of quantum superposition and the paradoxes that arise when quantum systems are scaled to the macroscopic world. At this thought experiment is a curious question: how can something be both alive and dead at the same time? 

Schrödinger’s Cat Experiment: A Quantum Mystery. 

The Thought Experiment: A Cat in a Box

In Schrödinger’s original thought experiment, imagine a cat is placed inside a sealed box, which contains the following elements:

1. A radioactive atom (a quantum system that has a 50% chance of decaying in a given time).
2. A Geiger counter to detect radiation.
3. A vial of poison.
4. A hammer connected to the Geiger counter that, if radiation is detected, will release the hammer and break the vial, killing the cat.

If the atom decays, the Geiger counter detects it, causing the hammer to break the poison vial and kill the cat. If the atom does not decay, the cat remains alive. The atom's decay is a quantum event, meaning it can exist in a superposition of decayed and undecayed states. But here’s where things get strange: according to quantum mechanics, until someone observes the system, the atom is in both states at once. As a result, the cat is theoretically both alive and dead at the same time.

This scenario creates a paradox when we think about how the quantum world (the behavior of the atom) and the macroscopic world (the fate of the cat) interact.

The Quantum Superposition

In quantum mechanics, particles like atoms exist in a superposition of all possible states until they are observed or measured. This idea is captured in Schrödinger's wave equation, which provides the mathematical description of the probability of finding a particle in a certain state.

Schrödinger's Equation (Time-Dependent):

$$i\hbar \frac{\partial}{\partial t} \Psi(x,t) = \hat{H} \Psi(x,t)$$

• $$Ψ(x,t): The wave function, which contains all possible information about the system.
• $\hat{H}$: The Hamiltonian operator, representing the total energy of the system.
• $\mathrm{<span\; style="color:\; \#274e13;\; font-family:\; arial;">\hslash :\; Reduced\; Planck’s\; constant.</span>}$
• $i$: The imaginary unit.

The wave function $\Psi(x,t)$ describes the quantum state of a system. Before measurement, the atom (and by extension, the cat) exists in a superposition of both decayed and undecayed states. The wave function collapses into a definite state (alive or dead) only when observed.

Cat Experiment. 

Copenhagen Interpretation: Observation and Collapse

One of the most widely accepted interpretations of quantum mechanics is the Copenhagen interpretation. According to this view, a system exists in superposition until it is observed. The act of measurement causes the wave function to "collapse" into one of the possible states. In the case of Schrödinger’s cat:

• Before opening the box, the cat is both alive and dead (superposition).
• Upon observation (when the box is opened), the wave function collapses, and the cat is either alive or dead.

This collapse represents the transition from the quantum world (where probabilities rule) to the classical world (where we experience definite outcomes).

The Many-Worlds Hypothesis

Another interpretation that seeks to resolve the Schrödinger's cat paradox is the Many-Worlds Interpretation. This theory suggests that every possible outcome of a quantum event actually happens, but in different parallel universes. In the case of the cat:

• In one universe, the cat is alive.
• In another universe, the cat is dead.

This interpretation eliminates the need for wave function collapse, as each possibility simply plays out in a separate universe.

Quantum Mechanics and the Measurement Problem

The Schrödinger’s Cat experiment highlights a central issue in quantum mechanics: the measurement problem. This problem deals with how and why observations cause a quantum system to collapse into a single state, transitioning from the probabilistic quantum world to the definite classical world.

Mathematical Expression for Superposition

To explain mathematically, the state of the cat can be represented as a superposition:

$$|\text{Cat}\rangle = \frac{1}{\sqrt{2}} \left( |\text{Alive}\rangle + |\text{Dead}\rangle \right)$$

Here, the cat is in a 50-50 superposition of being alive and dead. The probability of finding the cat in either state upon observation is 50%, but until the observation, the cat's true state is indeterminate.

Mathematical Representation

In quantum mechanics, the state of a system is described by a wave function, denoted by Ψ (psi). The wave function encodes all possible states of a system. In the case of the cat experiment, we can express the superposition of states mathematically as follows:

$$\Psi = \frac{1}{\sqrt{2}} \left( | \text{Alive} \rangle + | \text{Dead} \rangle \right)$$

Here, the cat is represented by the states $\mathrm{\mid }\text{Alive}⟩\; and\; \mid$$| \text{Dead} \rangle$, and the factor $\frac{1}{\sqrt{2}}$ ensures that the probabilities of both states sum to 1.

When the box is opened (the measurement is made), the wave function collapses into one of the two possible outcomes. The mathematical expression for the collapse is:

$$$$Ψcollapsed=∣Alive⟩orΨcollapsed=∣Dead⟩\Psi_{\text{collapsed}} = | \text{Alive} \rangle \quad \text{or} \quad \Psi_{\text{collapsed}} = | \text{Dead} \rangle

The probabilities are governed by the Born Rule, which states that the probability of an outcome is the square of the amplitude of the wave function for that state. For instance, if the cat has an equal chance of being alive or dead, the probabilities would be:

$$P(\text{Alive}) = |\langle \text{Alive} | \Psi \rangle|^2 = \frac{1}{2}, \quad P(\text{Dead}) = |\langle \text{Dead} | \Psi \rangle|^2 = \frac{1}{2}$$

This expresses that until the box is opened, both outcomes are equally likely.

The Role of Entanglement

Schrödinger’s Cat also illustrates the concept of quantum entanglement. The atom and the cat become entangled in such a way that the state of the cat is directly tied to the state of the atom:

• If the atom decays, the cat dies.
• If the atom does not decay, the cat remains alive.

Entanglement means the two systems (the atom and the cat) cannot be described independently of one another.

Fun and Curious Facts about Schrödinger’s Cat

1. Schrödinger’s Intention: Schrödinger originally devised this thought experiment to critique the Copenhagen interpretation, not to support it. He found the notion of a cat being both alive and dead absurd, using the thought experiment as a way to highlight the problems of applying quantum mechanics to everyday objects.

2. Applications to Quantum Computing: Schrödinger's cat has found a real-world application in quantum computing. The idea of superposition (being in multiple states at once) is at the heart of how quantum computers work, enabling them to perform complex calculations at unprecedented speeds.

3. Real-World Schrödinger’s Cats?: In recent years, scientists have been able to create real-world systems that mimic Schrödinger’s cat on a microscopic scale. They’ve used photons and other particles to show that quantum systems can indeed exist in superposition, though the "cat" in these experiments is far smaller and less complicated than a real animal.

4. Quantum Biology: Some scientists speculate that Schrödinger’s Cat may have applications in understanding quantum effects in biology, such as how plants use quantum mechanics in photosynthesis, where particles like electrons can exist in multiple places simultaneously.

Hypotheses and Interpretations Among Scientists

• Objective Collapse Theories: Some researchers propose that quantum systems naturally collapse into definite states after a certain amount of time or interaction with their environment. This avoids the need for observation to trigger the collapse.

• Quantum Darwinism: This hypothesis suggests that the classical world emerges through a process similar to natural selection, where certain quantum states are “selected” by their interactions with the environment, allowing them to become the definite states we observe.

Conclusion

Schrödinger’s Cat remains a profound symbol of the bizarre world of quantum mechanics. It reveals the strange and counterintuitive nature of the quantum realm, where particles can exist in multiple states, and observations change the nature of reality itself. While the cat is a thought experiment, its implications resonate throughout modern physics, from quantum computing to potential applications in quantum biology.

By exploring Schrödinger’s cat, we dive deeper into the mysteries of quantum superposition, measurement, and the transition from the microscopic quantum world to the macroscopic world we experience daily. The more we study, the more curious—and complex—this quantum world becomes. 

References:

1. Griffiths, D.J. (2004). Introduction to Quantum Mechanics. Pearson Prentice Hall.
2. Nielsen, M. A., & Chuang, I. L. (2010). Quantum Computation and Quantum Information. Cambridge University Press.
3. Everett, H. (1957). "Relative State Formulation of Quantum Mechanics." Reviews of Modern Physics.

These references will guide readers to explore the depth of quantum mechanics and its theoretical interpretations. 

"One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following diabolical device... one would, according to the Copenhagen interpretation, have to admit that the cat is both dead and alive at the same time."
— Erwin Schrödinger.