Patrick Reany
8 May 2024
Patrick:
The Schrodinger equation has a continuity equation for the conservation of probability, which is derived by a straightforward differentiation. Does this fact by itself tend to indicate that the Schrodinger equation must be a good approximation for modeling atomic physics, because the likelihood of getting probability conservation out of a false equation should be very small? Yes, I'm making a Bayesian argument, but it makes sense.
ChatGPT 3.5:
Your Bayesian argument has merit, as it considers the likelihood of obtaining a desired property (probability conservation) from an equation. The continuity equation derived from the Schrodinger equation indeed indicates that it maintains probability conservation, which is a fundamental aspect of quantum mechanics. This property is crucial for accurately modeling atomic and subatomic systems, where probabilities play a central role.
However, while the conservation of probability is a necessary condition for a valid physical model, it is not sufficient on its own to establish the accuracy or validity of the Schrodinger equation for all atomic physics phenomena. Other factors such as empirical evidence, agreement with experimental results, and consistency with other principles of physics also contribute to determining the suitability of the Schrodinger equation as a modeling tool.
In Bayesian terms, the fact that the Schrodinger equation yields a continuity equation for probability conservation can be viewed as providing evidence in favor of its validity for atomic physics modeling. But it's essential to consider this evidence within the broader context of empirical support and theoretical coherence.
Then I asked the same question to Copilot and it replied:
Copilot:
The continuity equation for the conservation of probability in quantum mechanics is indeed an important aspect of the Schrodinger equation. Let's delve into this topic further.
The probability conservation equation arises from the Schrodinger equation and ensures that the total probability associated with a quantum system remains constant over time. Here's how it works:
1. Probability Current:
- Start by considering the probability density function (wave function) \Psi(\mathbf{r}, t), which describes the probability of finding a particle at position \mathbf{r} at time t.
- The probability current is defined as:
\mathbf{J} = \frac{\hbar}{2m} \left(\Psi^* \nabla \Psi - \Psi \nabla \Psi^*\right)
where \hbar is the reduced Planck constant, m is the particle's mass, and \nabla represents the gradient operator.
- The probability current \mathbf{J} represents the flow of probability in space.
2. Conservation Equation:
- Now, let's consider the rate of change of probability within a volume V:
\frac{d}{dt} \int_V |\Psi|^2 dV = -\oint_S \mathbf{J} \cdot d\mathbf{S}
where S is the surface enclosing the volume V.
- The left-hand side represents the change in total probability within the volume, while the right-hand side represents the flux of probability across the surface.
- If we assume that the wave function satisfies the Schrodinger equation, we can show that the probability current satisfies the continuity equation:
\frac{\partial |\Psi|^2}{\partial t} + \nabla \cdot \mathbf{J} = 0
- This equation ensures that the total probability remains conserved over time.
3. Interpretation:
- The continuity equation implies that the probability density cannot accumulate or disappear within a region of space unless there are sources or sinks of probability (e.g., interactions with other particles).
- It also ensures that a normalized wave function remains normalized after time evolution.
4. Bayesian Argument:
- Your Bayesian argument is interesting! While the continuity equation itself doesn't directly prove the Schrodinger equation's accuracy, it does provide strong support.
- If the Schrodinger equation were fundamentally flawed, it would be unlikely to yield a probability conservation equation that consistently holds in various physical scenarios.
- However, keep in mind that the Schrodinger equation is an approximation, valid under specific conditions (e.g., non-relativistic, non-interacting particles). In more complex situations, relativistic effects and interactions may require more sophisticated equations.
In summary, the conservation of probability through the continuity equation reinforces the validity of the Schrodinger equation for modeling atomic and subatomic systems. While it doesn't guarantee perfection, it does suggest that the equation captures essential aspects of quantum behavior [1,2,4].
Source: Conversation with Bing, 5/8/2024
(1) Probability Conservation Equation - University of California, San Diego.
https://quantummechanics.ucsd.edu/ph130a/130_notes/node127.html.
(2) The Flux of Probability - University of California, San Diego.
https://quantummechanics.ucsd.edu/ph130a/130_notes/node122.html.
(3) Schrödinger equation - Wikipedia. https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation.
(4) Dynamical Exclusion of Probability and Energy Conservation. https://arxiv.org/pdf/2305.03141.pdf.
(5) en.wikipedia.org. https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation.