Foundations of Quantum Mechanics

by | Apr 23, 2025 | Quantum Mechanics | 0 comments

Thomas Lee Abshier
Author: Thomas Lee Abshier

The Foundations of QM
by Thomas Lee Abshier, ND
4/23/25

Thomas: What were the detectors used in the measurement of Quantum Mechanical phenomena in the early days, from the first experiments to the present.

AI: The early exploration of quantum mechanics (QM) relied heavily on indirect measurements and theoretical extrapolations, as direct detection of quantum states was not feasible with the technology of the time.

  1. Early Detectors in QM Research: In the early 20th century, experiments like the blackbody radiation studies by Max Planck and the photoelectric effect experiments by Albert Einstein used macroscopic detectors such as thermopiles, galvanometers, and photographic plates. These devices measured energy, light intensity, or electric current, indirectly revealing quantum phenomena.
  2. Detection of Spin States or Quantum Phenomena: The Stern-Gerlach experiment (1922) was a landmark in detecting quantum phenomena, specifically the quantization of angular momentum (spin). A beam of silver atoms was passed through a non-uniform magnetic field, and their deflection revealed discrete spin states. This was an indirect measurement, as the spin states were inferred from the spatial distribution of the atoms.
  3. Increase in Quantum Number (n): Observing changes in quantum numbers often involved spectroscopy. For example, the emission or absorption spectra of atoms provided evidence of transitions between quantized energy levels. These observations were interpreted using the Schrödinger Wave Equation (SWE) and the theoretical framework of quantum mechanics.
  4. Extrapolation from Theory: Much of the early understanding of QM came from theoretical insights. Planck’s quantization of energy in blackbody radiation and Bohr’s model of the hydrogen atom were based on mathematical formulations rather than direct detection of quantum states.
  5. Modern Detection of Quantum States: Today, advanced technologies like superconducting qubits, ion traps, and quantum dots allow for more direct manipulation and measurement of quantum states. Techniques such as quantum tomography and weak measurements provide insights into quantum states, but even now, many aspects of QM are inferred through indirect measurements and theoretical models.

In essence, the journey from early QM research to modern quantum technologies has been one of progressively refining our ability to detect, manipulate, and understand quantum states. Theoretical frameworks have always played a crucial role in interpreting experimental data and guiding the development of new detection methods.

Thomas: What are the modern detectors capable of directly detecting quantum states?

AI: Modern detectors capable of directly interacting with and measuring quantum states have advanced significantly. Here are some examples:

  1. Quantum Sensors: These include devices like nitrogen-vacancy (NV) centers in diamonds, which can detect minute variations in magnetic and electric fields. They are used in applications ranging from quantum computing to biological sensing.
  2. Superconducting Detectors: These are used in quantum computing and particle physics. They exploit quantum mechanical properties like superconductivity to measure quantum states with high precision.
  3. Trapped Ion Systems: These systems use ions confined in electromagnetic traps to study and manipulate quantum states. They are often used in quantum computing and fundamental physics research.
  4. Quantum Illumination Techniques: These methods use entangled photons to detect objects or measure distances with higher sensitivity than classical methods, even in noisy environments.

These technologies represent the cutting edge of quantum state detection, enabling researchers to probe and manipulate quantum phenomena with unprecedented precision. Let me know if you’d like to explore any of these in more detail!

Thomas: What is the physical reality represented by the wavefunction?

The wavefunction in quantum mechanics is a fundamental concept that encodes the quantum state of a system. It is indeed challenging to assign a concrete “physical” interpretation to the wavefunction itself, as it represents something that transcends the classical notions of particles and waves. Below is an explanation of what the wavefunction is, what it represents, and whether it has a physical counterpart.

1. What Is the Wavefunction?

The wavefunction, often denoted as

ψ(r,t)\psi(r, t)

is a complex-valued mathematical function that describes the quantum state of a system. It depends on the position r and time t, and its primary role is to encode all the information about the system.

Key properties of the wavefunction:

  • It is complex-valued, meaning it can be written as:
    ψ(r,t)=a(r,t)+ib(r,t),\psi(r, t) = a(r, t) + i b(r, t),

    where

    a(r,t)a(r, t)and

    b(r,t)b(r, t)are real functions, and

    iiis the imaginary unit

    i2=−1

  • The wavefunction itself is not directly observable. Instead, its magnitude squared,
    ∣ψ(r,t)∣2|\psi(r, t)|^2     

    gives the probability density of finding the particle at position r and time t.

    ∣ψ(r,t)∣2=Probability density at position r and time t.|\psi(r, t)|^2 = \text{Probability density at position } r \text{ and time } t.Thus, the wavefunction is a probability amplitude, which encodes both the likelihood of outcomes and the phase information required for quantum interference.

2. Is the Wavefunction Physical?

There is ongoing debate in the interpretation of quantum mechanics about whether the wavefunction represents something “real” or is merely a mathematical abstraction. Here are two broad perspectives:

a. The Wavefunction as a Physical Entity

Some interpretations suggest that the wavefunction has a real, physical existence. For example:

  • In the pilot wave theory (or de Broglie-Bohm theory), the wavefunction is treated as a real field that guides the motion of particles.
  • The wavefunction evolves deterministically according to the Schrödinger equation:
    iℏ∂ψ(r,t)∂t=H^ψ(r,t),i\hbar \frac{\partial \psi(r, t)}{\partial t} = \hat{H} \psi(r, t),

    is the reduced Planck’s constant, and

    H^\hat{H}is the Hamiltonian operator representing the system’s total energy.

b. The Wavefunction as an Abstract Tool

In other interpretations (e.g., the Copenhagen interpretation), the wavefunction is viewed as an abstract mathematical construct that represents our knowledge or predictions about a system, rather than a physical entity.

3. What Does the Wavefunction Represent?

While the wavefunction itself may not have a direct physical correlate, it encodes essential information about the system:

a. Probability Amplitudes

The wavefunction is a probability amplitude, meaning that its squared magnitude gives the probability density. For example:

  • In one dimension,
    ∣ψ(x,t)∣2 dx|\psi(x, t)|^2 \, dx     

    gives the probability of finding the particle between x and x + dx.

  • In three dimensions,
    ∣ψ(r,t)∣2 d3r|\psi(r, t)|^2 \, d^3r     

    gives the probability of finding the particle in a small volume d3rd^3raround position r.

b. Quantum Superposition

The wavefunction allows for superpositions of states, meaning a particle can exist in multiple states simultaneously. For instance:

 

ψ=c1ψ1+c2ψ2,\psi = c_1 \psi_1 + c_2 \psi_2,

c_1

and are complex coefficients. This superposition is the foundation of quantum phenomena like interference and entanglement.

c. Phase Information

The wavefunction also encodes phase information, which governs interference effects. Even though

∣ψ(r,t)∣2|\psi(r, t)|^2

gives probabilities, the relative phase between components of the wavefunction determines how they interfere.

4. Is There a Physical Correlate?

While the wavefunction itself might not correspond directly to any classical physical entity, its effects are observable. Here are ways in which the wavefunction manifests physically:

a. Probability Distributions

The most direct physical connection is through

∣ψ(r,t)∣2|\psi(r, t)|^2

which gives measurable probabilities. For example:

  • The interference pattern in the double-slit experiment corresponds to
    ∣ψ(r,t)∣2|\psi(r, t)|^2     

    demonstrating the wavefunction’s role in shaping observable phenomena.

b. Energy and Momentum

The wavefunction encodes information about a particle’s energy and momentum. For example:

  • The momentum operator acts on the wavefunction to give the particle’s momentum distribution:

     

    p^ψ=−iℏ∇ψ.

c. Observable Quantities

The wavefunction is used to calculate expectation values of physical observables. For an observable represented by an operator

O^\hat{O}

the expectation value is:

⟨O⟩=∫ψ∗(r,t)O^ψ(r,t) d3r.\langle O \rangle = \int \psi^*(r, t) \hat{O} \psi(r, t) \, d^3r. 

5. The Connection Between Math and Reality

As you noted, the mathematics of the wavefunction appears abstract, but its ability to predict experimental outcomes suggests a profound connection between the math and the underlying physical reality. This raises philosophical questions, such as:

  • Does the wavefunction describe reality directly, or is it simply a tool for organizing our observations?
  • Why does abstract mathematics correspond so well to the behavior of the physical world?

These questions remain open, but the success of quantum mechanics suggests that the wavefunction reflects essential truths about the nature of reality.

6. Summary: A Balanced Perspective

The wavefunction is a mathematical representation that encodes the quantum state of a system. It is not directly observable, but it describes the behavior of underlying physical processes. Key points include:

  1. The wavefunction
    ψ(r,t)\psi(r, t)     

    is a complex-valued function that encodes probability amplitudes and phase information.

  2. The squared magnitude
    ∣ψ(r,t)∣2|\psi(r, t)|^2     

    gives the probability density of finding a particle at position r and time t.

  3. While the wavefunction itself may be abstract, it predicts concrete, observable phenomena, such as interference patterns and energy quantization.

Your insight is correct: the wavefunction is not “just math.” It describes or corresponds to some aspect of the physical reality underlying quantum phenomena. Even if its exact nature remains elusive, its effects are directly observable and indispensable for understanding the quantum world.

Thomas: We can say for certain that there is a physical substance/entity/something underneath all quantum phenomena. That is, quantum phenomena do not arise from mathematical expressions, nor out of space with nothing in it. Thus, we can state with certainty the wavefunction describes the behavior/state or some aspect of some physical reality. The math is abstract, but the fact that it predicts behavior means that the math describes the behavior of something. Apparently when that abstract mathematical function is manipulated in various ways, it reveals more concrete physically recognized entities such as the probability of finding a particle at a location. Would this be accurate?

AI: Yes, your formulation is accurate and reflects a nuanced understanding of the wavefunction and the relationship between mathematical models and physical reality. While the wavefunction itself is a mathematical construct, its predictive success strongly suggests that it describes or corresponds to something real and physical, even if we do not yet fully understand the nature of that “something.” Let me expand on the key points of your explanation and connect them to broader discussions in physics.

1. The Wavefunction Describes an Underlying Physical Reality

  • As you correctly stated, quantum phenomena do not arise from mathematical expressions alone. The wavefunction is not “just math”; it is a tool that encodes and predicts the behavior of physical systems. This implies that the wavefunction corresponds to some aspect of an underlying physical reality.
  • The fact that the wavefunction can be manipulated mathematically to yield concrete, testable predictions (e.g., probabilities of particle detection, energy levels, etc.) suggests that it captures something essential about the behavior/state of the physical world.

This aligns with the view that the wavefunction is a representation of the state of a quantum system, even if the exact nature of what it represents remains an open question.

2. The Abstract Nature of the Wavefunction

The abstract nature of the wavefunction is both a strength and a challenge:

  • The wavefunction is expressed in Hilbert space, a high-dimensional mathematical space that is not directly observable. This abstraction allows it to encode a vast amount of information about the system, including probabilities, superpositions, and interference effects.
  • However, this abstraction makes it difficult to identify a direct, one-to-one relationship between the wavefunction and familiar classical concepts like “particles” or “fields.”

Despite this, observable phenomena (e.g., interference patterns, energy quantization) emerge from the wavefunction, strongly indicating that it reflects something real and physical, even if it does so indirectly.

3. The Wavefunction Encodes Physical Behavior

As you pointed out, manipulating the wavefunction mathematically reveals physical quantities such as:

  • The probability density (
    ∣ψ(r,t)∣2|\psi(r, t)|^2     

    of finding a particle at a specific location.

  • The expectation values of measurable quantities (e.g., position, momentum, energy), calculated using operators acting on the wavefunction.

This ability to extract meaningful physical information from the wavefunction implies that it describes the behavior or state of the underlying physical entity responsible for quantum phenomena.

4. The Nature of the “Something” Underlying Quantum Phenomena

While we can infer that there is some physical entity or structure underlying quantum phenomena, its exact nature remains mysterious. Here are some possibilities:

a. The Wavefunction as a Physical Field

One interpretation is that the wavefunction itself is a real physical field, similar to the electromagnetic field, but existing in a more abstract space (e.g., configuration space or Hilbert space). This view is supported by the fact that the wavefunction evolves deterministically according to the Schrödinger equation (until measurement occurs).

b. The Wavefunction as a Representation of Probability

Another perspective is that the wavefunction represents probability amplitudes, which are related to our knowledge or predictions about the system rather than the system itself. In this view, the wavefunction is not “real” in the same way as particles or fields but is instead a mathematical tool for predicting the outcomes of measurements.

c. The Wavefunction as a Guide

In the pilot wave (de Broglie-Bohm) theory, the wavefunction is a real entity that guides the motion of particles, which always have definite positions and trajectories. This interpretation gives the wavefunction a direct physical role.

d. Emergent Reality

Some approaches suggest that the wavefunction reflects an emergent property of a deeper underlying reality. For example, quantum phenomena might arise from interactions within a more fundamental structure (e.g., spacetime, information, or something yet unknown).

5. The Connection Between Math and Reality

As you noted, the mathematics of the wavefunction appears abstract, but its ability to predict concrete experimental outcomes suggests a deep connection between math and reality. This connection raises profound philosophical questions, such as:

  • Does the wavefunction describe reality directly, or is it a tool for organizing our observations?
  • Why does abstract mathematics correspond so well to the behavior of the physical world?

These questions remain open, but the success of quantum mechanics suggests that the wavefunction encodes a profound truth about the nature of reality.

6. Summary: A Balanced Perspective

Your description captures the essence of current thinking about the wavefunction:

  • While the wavefunction is a mathematical abstraction, it describes the behavior or state of some underlying physical entity or structure responsible for quantum phenomena.
  • The abstract nature of the wavefunction does not diminish its physical relevance; instead, it reflects the complexity of quantum reality, which defies classical intuition.
  • The wavefunction’s ability to predict observable phenomena, such as probabilities and energy levels, confirms that it is deeply connected to the physical world, even if its exact nature remains mysterious.

In this sense, the wavefunction is not “just math” but a powerful representation of the underlying reality that quantum mechanics seeks to describe. It serves as a bridge between the abstract mathematical framework of quantum theory and the concrete physical phenomena we observe.

Thomas: What is the Shrodinger Wave Equation?

AI: 

What is the Schrödinger Wave Equation?

The Schrödinger wave equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system evolves over time. It is central to understanding the behavior of particles, such as electrons, at the atomic and subatomic levels. There are two primary forms of the Schrödinger wave equation: the time-dependent form and the time-independent form.

1. The Time-Dependent Schrödinger Equation

The time-dependent Schrödinger equation is given by:

 

iℏ∂ψ(r,t)∂t=H^ψ(r,t),  

where:


  • ψ(r,t)\psi(r, t)     

    is the wavefunction, which describes the quantum state of the system as a function of position rr 

    and time tt 

    .


  • ii     

    is the imaginary unit ( i2=−1i^2 = -1 

    ).


  • ℏ\hbar     

    is the reduced Planck’s constant ( ℏ=h/2π\hbar = h / 2\pi 

    ,


  • h≈6.626×10−34       

  • ∂ψ(r,t)∂t\frac{\partial \psi(r, t)}{\partial t}     

    is the partial derivative of ψ(r,t)\psi(r, t) 

    with respect to time.


  • H^\hat{H}     

    is the Hamiltonian operator, which represents the total energy of the system and is typically expressed as:
    H^=−ℏ22m∇2+V(r,t).\hat{H} = -\frac{\hbar^2}{2m} \nabla^2 + V(r, t). 

    What is the Schrödinger Wave Equation?

    The Schrödinger wave equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system evolves over time. It is central to understanding the behavior of particles, such as electrons, at the atomic and subatomic levels. There are two primary forms of the Schrödinger wave equation: the time-dependent form and the time-independent form.

    1. The Time-Dependent Schrödinger Equation

    The time-dependent Schrödinger equation is given by:

     

    iℏ∂ψ(r,t)∂t=H^ψ(r,t),i\hbar \frac{\partial \psi(r, t)}{\partial t} = \hat{H} \psi(r, t),

    where:


    • ψ(r,t)\psi(r, t)       

      is the wavefunction, which describes the quantum state of the system as a function of position rr 

      and time tt 

      .


    • ii       

      is the imaginary unit ( i2=−1i^2 = -1 

      ).


    • ℏ\hbar       

      is the reduced Planck’s constant ( ℏ=h/2π\hbar = h / 2\pi 

      ,

    • where
      h≈6.626×10−34 J-       

      .


    • ∂ψ(r,t)∂t\frac{\partial \psi(r, t)}{\partial t}       

      is the partial derivative of ψ(r,t)\psi(r, t) 

      with respect to time.


    • H^\hat{H}       

      is the Hamiltonian operator, which represents the total energy of the system and is typically expressed as:
      H^=−ℏ22m∇2+V(r,t).\hat{H} = -\frac{\hbar^2}{2m} \nabla^2 + V(r, t). 


      Here:


      • −ℏ22m∇2-\frac{\hbar^2}{2m} \nabla^2         

        is the kinetic energy operator ( mm 

        is the mass of the particle, and ∇2\nabla^2 

        is the Laplacian operator).


      • V(r,t)V(r, t)         

        is the potential energy of the system, which depends on position rr 

        and possibly time tt 

        .

    The time-dependent Schrödinger equation governs the evolution of the wavefunction over time and is used to describe systems where time-dependent phenomena are important.

    2. The Time-Independent Schrödinger Equation

    For systems where the potential energy

    V(r)V(r) does not depend on time, the time-dependent Schrödinger equation reduces to the time-independent Schrödinger equation:

     

    H^ψ(r)=Eψ(r),\hat{H} \psi(r) = E \psi(r),

    where:


    • ψ(r)\psi(r)       

      is the wavefunction that depends only on position r  


    • EE       

      is the total energy of the system (a constant).

    The time-independent Schrödinger equation is often used to solve problems involving stationary states, such as the energy levels of an electron in an atom.

    3. Physical Interpretation

    The wavefunction

    ψ(r,t)\psi(r, t)does not correspond directly to something physically measurable. However, its squared magnitude,

    ∣ψ(r,t)∣2|\psi(r, t)|^2, represents the probability density of finding the particle at position

    rrat time

    tt. This probabilistic interpretation is one of the key features of quantum mechanics.

    4. Applications

    The Schrödinger wave equation has numerous applications, including:

    1. Particle in a Box: Describing a particle confined to a finite region of space.
    2. Quantum Harmonic Oscillator: Modeling systems with restoring forces, such as vibrating molecules.
    3. Hydrogen Atom: Solving for the energy levels and orbitals of an electron in a hydrogen atom.
    4. Quantum Tunneling: Explaining how particles can pass through potential barriers even if they lack the classical energy to do so.

    Summary

    The Schrödinger wave equation is a cornerstone of quantum mechanics, providing a mathematical framework to describe the quantum state of a system. It exists in two forms:

    • The time-dependent Schrödinger equation, which describes how the wavefunction evolves over time.
    • The time-independent Schrödinger equation, which is used for stationary states.

    Solving the Schrödinger equation can determine the behavior of quantum systems and predict phenomena that are often counterintuitive in the classical world.

    Here:


    • −ℏ22m∇2-\frac{\hbar^2}{2m} \nabla^2       

      is the kinetic energy operator ( mm 

      is the mass of the particle, and ∇2\nabla^2 

      is the Laplacian operator).


    • V(r,t)V(r, t)       

      is the potential energy of the system, which depends on position rr 

      and possibly time tt 

      .

The time-dependent Schrödinger equation governs the evolution of the wavefunction over time and is used to describe systems where time-dependent phenomena are important.

2. The Time-Independent Schrödinger Equation

For systems where the potential energy

V(r)V(r)does not depend on time, the time-dependent Schrödinger equation reduces to the time-independent Schrödinger equation:

 

H^ψ(r)=Eψ(r),\hat{H} \psi(r) = E \psi(r),

where:


  • ψ(r)\psi(r)     

    is the wavefunction that depends only on position r 


  •  

    EE 

    is the total energy of the system (a constant).

The time-independent Schrödinger equation is often used to solve problems involving stationary states, such as the energy levels of an electron in an atom.

3. Physical Interpretation

The wavefunction

ψ(r,t)\psi(r, t)does not correspond directly to something physically measurable. However, its squared magnitude,

∣ψ(r,t)∣2|\psi(r, t)|^2, represents the probability density of finding the particle at position

rrat time

tt. This probabilistic interpretation is one of the key features of quantum mechanics.

4. Applications

The Schrödinger wave equation has numerous applications, including:

  1. Particle in a Box: Describing a particle confined to a finite region of space.
  2. Quantum Harmonic Oscillator: Modeling systems with restoring forces, such as vibrating molecules.
  3. Hydrogen Atom: Solving for the energy levels and orbitals of an electron in a hydrogen atom.
  4. Quantum Tunneling: Explaining how particles can pass through potential barriers even if they lack the classical energy to do so.

Summary

The Schrödinger wave equation is a cornerstone of quantum mechanics, providing a mathematical framework to describe the quantum state of a system. It exists in two forms:

  • The time-dependent Schrödinger equation, which describes how the wavefunction evolves over time.
  • The time-independent Schrödinger equation, which is used for stationary states.

Solving the Schrödinger equation can determine the behavior of quantum systems and predict phenomena that are often counterintuitive in the classical world.

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