FrontierScience / 69bd11e8-5f96-45dc-8123-b9d800101430 / Domain Wall 2**

**Domain Wall 2**

Consider a one-dimensional model where the magnetization as a function of the `x`-coordinate is denoted by `M(x)`. The energy of the system is

`E=∫_(−∞)^(∞)dx[(ρ)/(2)((dM)/(dx))²+(κ)/(4)(M²−M₀²)²]`

Here, `ρ>0`, `κ>0`, and `M₀` are constants related to the kinetic term and the shapes of the potential respectively.

Suppose that the boundary condition is

`M(∞)=−M(−∞)=M₀`

which means the magnetization should gradually change from `−M₀` to `M₀` somewhere in space.
At the temperature `T=0K`,  Determine the expectation value of the energy `⟨E⟩` in terms of the parameters of the model.

Think step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.

**Domain Wall 2**

Consider a one-dimensional model where the magnetization as a function of the `\(x\)`-coordinate is denoted by `\(M(x)\)`. The energy of the system is

`\(E=\int_{-\infty}^{\infty}dx{\left[\frac\rho2{\left(\frac{dM}{dx}\right)}^2+\frac\kappa4{\left(M^2-M_0^2\right)}^2\right]}\)`

Here, `\(\rho>0\)`, `\(\kappa>0\)`, and `\(M_0\)` are constants related to the kinetic term and the shapes of the potential respectively.

Suppose that the boundary condition is

`\(M(\infty)=-M(-\infty)=M_0\)`

which means the magnetization should gradually change from `\(-M_0\)` to `\(M_0\)` somewhere in space.
At the temperature `\(T=0K\)`,  Determine the expectation value of the energy `\(\langle E\rangle\)` in terms of the parameters of the model.

Think step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.