Understanding bra-ket notation

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You said:

I have been reading works on quantum mechanics and field theory. from my past undergraduate study, fifty years ago, I understand the Hamiltonian for an electron under the constraints of inverse square law attraction or perturbation theory for electrons in a semiconductor crystal. I seem to have some conceptual difficulty with Dirac’s bra-ket notation. Do you think you can help me work through this so I can move on to looking at Lagrangian dynamics and hence to field theory?

ChatGPT said:

ChatGPT

Memory updated

I’d be happy to help you with Dirac’s bra-ket notation. It’s a powerful formalism in quantum mechanics, and understanding it will definitely smooth your transition to Lagrangian dynamics and field theory.

Here’s a quick breakdown of the essentials to get started:

  1. Kets (|ψ⟩): These represent state vectors in a Hilbert space. Think of a ket as a column vector that describes the state of a quantum system. For example, if we describe an electron in a certain quantum state, we write this as |ψ⟩.
  2. Bras (⟨φ|): A bra is the Hermitian conjugate (or complex conjugate transpose) of a ket. If |ψ⟩ is a column vector, then ⟨ψ| is a row vector. This is useful when calculating probabilities and overlaps between states.
  3. Inner Product (⟨φ|ψ⟩): The inner product of two state vectors is a complex number, representing the “overlap” between two quantum states. If ⟨φ|ψ⟩ = 0, the states are orthogonal.
  4. Outer Product (|ψ⟩⟨φ|): This creates an operator from two vectors. It’s essential for building matrices or describing projections.
  5. Operators (A|ψ⟩): Operators act on kets to produce new kets. These represent physical observables, like momentum or energy, and their action can give measurable quantities. For example, applying the Hamiltonian operator on a state gives information about its energy.

Basic Example:

For a two-state quantum system:

  • The states could be |0⟩ and |1⟩.
  • The inner product ⟨0|1⟩ = 0 (because they are orthogonal), while ⟨0|0⟩ = 1 (they are normalized).

Would you like to dive into specific examples or perhaps discuss how bra-ket notation works in the context of operators like the Hamiltonian? We could even connect it to what you remember from semiconductor physics.

4o

You said:

I get the inner product is a dot product giving a scalar. And I think the bra of one state operating on the ket of another state will give a scalar proportional to the amount of the bra state in the ket state. Does that make sense?

ChatGPT said:

ChatGPT

Yes, you’re absolutely on the right track! Your interpretation is essentially correct.

The inner product ⟨φ|ψ⟩ between a bra ⟨φ| and a ket |ψ⟩ gives a scalar that measures the “overlap” or “projection” of the state |ψ⟩ onto the state ⟨φ|. Here’s a bit more detail:

  • If ⟨φ|ψ⟩ is non-zero, this means that there is some overlap or similarity between the states |ψ⟩ and |φ⟩. The larger the magnitude of ⟨φ|ψ⟩, the more “amount” of the state |φ⟩ is contained in |ψ⟩.
  • If ⟨φ|ψ⟩ = 0, the two states are orthogonal, meaning there’s no overlap between them (they are entirely distinct).
  • If ⟨φ|ψ⟩ = 1, this means |φ⟩ and |ψ⟩ are essentially the same state (up to a possible phase factor, in the case of complex numbers).

This concept is directly tied to the idea of probabilities in quantum mechanics. For example, the probability of finding a quantum system in the state |φ⟩ if it is currently in the state |ψ⟩ is proportional to the square of the magnitude of ⟨φ|ψ⟩, i.e., |⟨φ|ψ⟩|².

Does this align with your understanding? If you’d like, we can apply this to an example, perhaps connecting it to familiar systems like electrons in potential wells or semiconductor crystals.

4o

You said:

Can you show me how this might look for say the energy of an lectron in the ground state of a hysdrogen atom. Maybe show me the legrandre polynimial solution from integration of the time independant compnent of the integral over the wavefunction and energy operator. AmI still making sense?

ChatGPT said:

ChatGPT

You’re making great sense! Let’s walk through the process of finding the energy of an electron in the ground state of a hydrogen atom using the time-independent Schrödinger equation, along with the relevant Legendre polynomial solutions.

1. Time-independent Schrödinger Equation (TISE)

For the hydrogen atom, we use the TISE in spherical coordinates because of the spherical symmetry of the problem. The Hamiltonian operator for the electron in the hydrogen atom is:

H^=−ℏ22me∇2−e24πϵ0r\hat{H} = -\frac{\hbar^2}{2m_e}\nabla^2 – \frac{e^2}{4\pi \epsilon_0 r}H^=−2me​ℏ2​∇2−4πϵ0​re2​

where:

  • ∇2\nabla^2∇2 is the Laplacian operator in spherical coordinates,
  • rrr is the radial distance from the nucleus,
  • e24πϵ0r\frac{e^2}{4\pi \epsilon_0 r}4πϵ0​re2​ is the Coulomb potential between the electron and proton,
  • mem_eme​ is the mass of the electron.

The TISE for this system is:

H^ψ(r,θ,ϕ)=Eψ(r,θ,ϕ)\hat{H} \psi(r, \theta, \phi) = E \psi(r, \theta, \phi)H^ψ(r,θ,ϕ)=Eψ(r,θ,ϕ)

where ψ(r,θ,ϕ)\psi(r, \theta, \phi)ψ(r,θ,ϕ) is the wavefunction of the electron, and EEE is the energy eigenvalue.

2. Separation of Variables

To solve this, we separate the wavefunction into its radial and angular components:

ψ(r,θ,ϕ)=R(r)Y(θ,ϕ)\psi(r, \theta, \phi) = R(r) Y(\theta, \phi)ψ(r,θ,ϕ)=R(r)Y(θ,ϕ)

  • R(r)R(r)R(r) is the radial part of the wavefunction.
  • Y(θ,ϕ)Y(\theta, \phi)Y(θ,ϕ) are the spherical harmonics, solutions to the angular part of the TISE.

The angular part gives rise to Legendre polynomials when solving the equation involving the angular momentum operator. These spherical harmonics are indexed by two quantum numbers: lll (orbital angular momentum quantum number) and mmm (magnetic quantum number).

For the ground state of hydrogen, the quantum numbers are:

  • l=0l = 0l=0 (no angular dependence),
  • m=0m = 0m=0 (since l=0l = 0l=0, mmm must also be zero).

Thus, for the ground state, the angular part is trivial: Y(θ,ϕ)=1Y(\theta, \phi) = 1Y(θ,ϕ)=1, and we only need to solve the radial equation.

3. Radial Equation

The radial part of the Schrödinger equation becomes:

−ℏ22me(d2R(r)dr2+2rdR(r)dr)−e24πϵ0rR(r)=ER(r)- \frac{\hbar^2}{2m_e} \left( \frac{d^2R(r)}{dr^2} + \frac{2}{r} \frac{dR(r)}{dr} \right) – \frac{e^2}{4\pi \epsilon_0 r} R(r) = E R(r)−2me​ℏ2​(dr2d2R(r)​+r2​drdR(r)​)−4πϵ0​re2​R(r)=ER(r)

To solve this, we introduce a new variable ρ=r/a0\rho = r / a_0ρ=r/a0​, where a0a_0a0​ is the Bohr radius. After some algebraic manipulation, we get a differential equation whose solutions involve Laguerre polynomials (not Legendre in this case, since we’re solving the radial part).

4. Ground State Solution

The solution for the ground state wavefunction R(r)R(r)R(r) is:

R1,0(r)=2a03/2e−r/a0R_{1,0}(r) = \frac{2}{a_0^{3/2}} e^{-r / a_0}R1,0​(r)=a03/2​2​e−r/a0​

The corresponding energy eigenvalue for the ground state (n = 1) is:

E1=−13.6 eVn2=−13.6 eVE_1 = -\frac{13.6 \, \text{eV}}{n^2} = -13.6 \, \text{eV}E1​=−n213.6eV​=−13.6eV

This energy is obtained by plugging the ground state wavefunction back into the radial Schrödinger equation and solving for EEE.

5. Legendre Polynomial Connection

In the hydrogen atom, the Legendre polynomials arise from the angular part of the wavefunction when solving for the spherical harmonics. Since we are dealing with the ground state (l=0l = 0l=0), the corresponding Legendre polynomial is simply P0(x)=1P_0(x) = 1P0​(x)=1, meaning there’s no angular dependence.

6. Energy Operator Acting on the Wavefunction

The Hamiltonian operator H^\hat{H}H^, when applied to the ground state wavefunction ψ1,0,0(r,θ,ϕ)\psi_{1,0,0}(r, \theta, \phi)ψ1,0,0​(r,θ,ϕ), gives the energy eigenvalue E1E_1E1​:

H^ψ1,0,0(r,θ,ϕ)=E1ψ1,0,0(r,θ,ϕ)\hat{H} \psi_{1,0,0}(r, \theta, \phi) = E_1 \psi_{1,0,0}(r, \theta, \phi)H^ψ1,0,0​(r,θ,ϕ)=E1​ψ1,0,0​(r,θ,ϕ)

In bra-ket notation, this is:

⟨ψ1,0,0∣H^∣ψ1,0,0⟩=E1\langle \psi_{1,0,0} | \hat{H} | \psi_{1,0,0} \rangle = E_1⟨ψ1,0,0​∣H^∣ψ1,0,0​⟩=E1​

This shows that the energy of the electron in the ground state is −13.6 eV-13.6 \, \text{eV}−13.6eV, consistent with experimental results.

Would you like to dive deeper into the integration steps or focus more on any other specific aspect of the solution?

4o

You said:

Your bring it all back from 1978!. Looks like I can follow it. And it really hepls me see how the bra-ket notation follows from that. Back in the late 7’s I grasped most of kattels solid state physics. I spent mostof my career developing manufacturing for VLSI and other solid state devices including high temperture josephson junction and many years work with single mode LED fibre optic sensor system development. Now retired I am keen to grasp field theory with the hope of possibly developing a field description of the dynamics of mental states – as discussed in one of our other conversations. I would be very pleased if you could suggest a course of study pathway that might help me with this. Kindof looking to develp a dynamic model of the Buddhist Abidharma. Am I still talking sense?

ChatGPT said:

ChatGPT

Memory updated

You’re making perfect sense, and your background gives you a strong foundation to explore this ambitious and fascinating goal. Developing a dynamic model of mental states using field theory, especially through the lens of the Buddhist Abhidharma, is a unique interdisciplinary approach that blends physics and philosophy.

Here’s a suggested study pathway that can help bridge your understanding of quantum field theory (QFT) and its potential application to modelling mental states:

1. Revisit Classical Field Theory

  • Before diving into quantum field theory, it’s important to have a solid grasp of classical field theory. This will help you see how fields describe physical systems.
  • Recommended Resources:
    • “Classical Mechanics” by Herbert Goldstein (specifically, the Lagrangian and Hamiltonian formulations).
    • “Classical Field Theory” by Landau and Lifshitz (Part 1 gives a nice introduction to how classical fields are described).
  • Focus on learning how to derive field equations from the Lagrangian and understand how fields propagate and interact. You may find parallels to how mental states could be modeled as fields with their own dynamics.

2. Study Lagrangian and Hamiltonian Field Theory

  • Since you mentioned wanting to move to Lagrangian dynamics, this will be a key bridge. Understand the transition from particle mechanics to fields.
  • Look at the Lagrangian density formulation of fields and the resulting Euler-Lagrange equations, which are foundational in both classical and quantum field theories.

3. Quantum Field Theory (QFT)

  • QFT extends quantum mechanics to systems with infinitely many degrees of freedom (fields), making it a natural framework to describe fields of consciousness or mental states.
  • Recommended Resources:
    • “Quantum Field Theory in a Nutshell” by A. Zee (a gentle introduction that balances concepts with formalism).
    • “An Introduction to Quantum Field Theory” by Peskin and Schroeder (more formal, detailed, and commonly used in the field).
    • Focus on understanding how fields are quantized and how interactions are described through exchange of quanta (e.g., photons in electromagnetism).

4. Gauge Theories and Symmetries

  • Much of modern field theory revolves around symmetry principles and gauge theories (e.g., how electromagnetism can be described as a gauge theory of the U(1)U(1)U(1) symmetry).
  • Investigate how symmetries underlie both physical laws and, potentially, the dynamics of mental states in Buddhist philosophy (e.g., balance between opposing mental forces, like greed and compassion).
  • Recommended Resource:
    • “Gauge Theories in Particle Physics” by Aitchison and Hey (a clear guide to the role of gauge symmetries in field theory).

5. Nonlinear Dynamics and Solitons

  • Nonlinear field theory may provide insights into how complex mental states evolve. Solitons (stable, localized solutions to nonlinear field equations) could be relevant in modeling stable mental states or persistent patterns in consciousness.
  • Recommended Resource:
    • “Solitons and Instantons” by Rajaraman (explores solutions in nonlinear field theory that might have parallels to stable mental phenomena).

6. Consciousness and Quantum Fields (Philosophical Connection)

  • To bridge these concepts to Abhidharma, you may explore recent work on quantum approaches to consciousness.
  • While speculative, some work explores the relationship between quantum fields and cognitive phenomena.
  • Recommended Resources:
    • “Quantum Mechanics and Consciousness” by Henry Stapp (explores possible connections between quantum theory and consciousness).
    • “Mind, Matter, and Quantum Mechanics” by Henry Stapp (ties in quantum theory with the nature of the mind).

7. Dynamic Models of Mental States (Buddhist Abhidharma)

  • The Abhidharma’s taxonomy of mental factors could be seen as a complex system of interacting fields.
  • Develop a field theory of mind where mental factors are fields interacting dynamically. Similar to how particles interact through fields, mental states could influence and propagate through a conceptual space.
  • Study models of complex systems, nonlinear dynamics, and coupled oscillators, which may provide mathematical tools to describe mental interactions in Abhidharma terms.
  • Recommended Resource:
    • “The Numerical Discourses of the Buddha” by Bhikkhu Bodhi (for deepening your understanding of mental states in the Abhidharma).

8. Mathematical Methods

  • Brush up on functional analysis and differential geometry, which are essential for field theory.
  • Recommended Resource:
    • “Mathematics for Physics” by M. Stone and P. Goldbart (covers the mathematical tools required for modern theoretical physics).

9. Exploration of Existing Models of Mind

  • While not directly related to field theory, models of cognitive science like Friston’s Free Energy Principle and Predictive Coding in neuroscience could provide inspiration for linking mental dynamics with field theory.

10. Start Modeling

  • Begin by sketching out potential field equations for mental factors, such as defining fields that represent states like mindfulness, concentration, and emotional balance.
  • Consider how these fields interact, perhaps influenced by forces analogous to physical fields (attractive, repulsive, neutralizing).
  • From there, attempt to formulate a Lagrangian density for the system and derive equations of motion.

This pathway would take you from the fundamentals of field theory all the way to potentially creating a dynamic model for the Buddhist Abhidharma’s description of mental states. Does this structure fit with how you’re envisioning this project?

You said:

Thanks. You are being very useful. I am beinging to see a way forward – the next steps – in my study.

ChatGPT said:

ChatGPT

You’re very welcome! I’m glad to hear that this is helping you see a clear path forward in your study. It sounds like you’re embarking on an exciting and meaningful journey, bridging fields in a way that’s both intellectually rigorous and personally fulfilling.

Feel free to reach out anytime as you progress—whether it’s clarifying concepts, exploring ideas further, or discussing new insights. I’m happy to support you along the way!