Confused about dot product step while deriving the Liouville equation

Thread starter oristo42
Start date Oct 20, 2017
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Confused deriving Dot Dot product Product

In summary, the dot product, also known as the scalar product, is a mathematical operation used to calculate the rate of change of a system's density with respect to time in the context of the Liouville equation. It is important in deriving the equation as it relates the time derivative of density to the Hamiltonian and helps understand probability flow in phase space. The dot product differs from the cross product in its result (scalar vs vector) and its measurement (similarity vs perpendicularity). In the Liouville equation, the dot product is performed by multiplying and summing the components of the gradient of density and the Hamiltonian vector. An example of using the dot product in the Liouville equation is calculating the time derivative of the system

Oct 20, 2017

oristo42

So, while textbook was deriving Liouville eq, this is one of step the book uses. I don't understand this step at all. Why is this step true?

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Oct 20, 2017

jedishrfu

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I think its the product rule for the del operator:

The power of the del notation is shown by the following product rule:

$6890a5ba5695e2dd567190fa3f5288abf756cc5d$

https://en.wikipedia.org/wiki/Del

1. What is the dot product in the context of the Liouville equation?

The dot product, also known as the scalar product, is a mathematical operation that takes two vectors and produces a scalar value. In the context of the Liouville equation, it is used to calculate the rate of change of a system's density with respect to time.

2. Why is the dot product important in deriving the Liouville equation?

The dot product is important because it allows us to express the time derivative of a system's density in terms of the Hamiltonian, which is a key component of the Liouville equation. It also helps us to understand the flow of probability in phase space.

3. What is the difference between the dot product and the cross product?

The dot product and the cross product are both mathematical operations involving vectors, but they have different results. The dot product produces a scalar value, while the cross product produces a vector. Additionally, the dot product measures the similarity between two vectors, while the cross product measures their perpendicularity.

4. How do you perform the dot product in the Liouville equation?

In the context of the Liouville equation, the dot product is performed by multiplying the components of two vectors and then summing the results. Specifically, we take the dot product of the gradient of the system's density (represented by the del operator) and the Hamiltonian vector.

5. Can you provide an example of using the dot product in the Liouville equation?

Yes, let's say we have a system with a density function ρ(x,y,z) and a Hamiltonian function H(x,y,z). To calculate the time derivative of the system's density, we would take the dot product of the gradient of ρ (represented by ∇ρ) and the Hamiltonian vector (represented by ∇H). This can be expressed as dρ/dt = ∇ρ ⋅ ∇H.