Merge branch 'contentupdate' into 'master'

Correction of the description in the Connection Formulas

See merge request !11

src/wkb_bound.md

@@ -76,15 +76,15 @@ Below, we will see that the boundary conditions *(connection formulas)* impose c
     === "Positive slope"
         | $E > V(x)$ | $E < V(x)$ |
         | :-----------: | :-----------: |
-        | $$\frac{2C}{\sqrt{p(x)}} \sin\left( \frac{1}{\hbar} \int_x^{x_t} p(x') dx' + \frac{\pi}{4} \right)$$ | $$\frac{C}{\sqrt{\|p(x)\|}} e^{-\frac{1}{\hbar} \int_{x_t}^x \|p(x')\|dx'} \quad \text{ (decaying)}$$ |
-        | $$\frac{D}{\sqrt{p(x)}} \cos\left( \frac{1}{\hbar} \int_x^{x_t} p(x') dx' + \frac{\pi}{4} \right)$$ | $$\frac{D}{\sqrt{\|p(x)\|}} e^{\frac{1}{\hbar} \int_{x_t}^x \|p(x')\|dx'} \quad \text{ (blowing up)}$$ |
+        | $$\frac{2C}{\sqrt{p(x)}} \sin\left( \frac{1}{\hbar} \int_x^{x_t} p(x') dx' + \frac{\pi}{4} \right)$$ | $$\frac{C}{\sqrt{\|p(x)\|}} e^{-\frac{1}{\hbar} \int_{x_t}^x \|p(x')\|dx'} \quad \text{ (blowing up)}$$ |
+        | $$\frac{D}{\sqrt{p(x)}} \cos\left( \frac{1}{\hbar} \int_x^{x_t} p(x') dx' + \frac{\pi}{4} \right)$$ | $$\frac{D}{\sqrt{\|p(x)\|}} e^{\frac{1}{\hbar} \int_{x_t}^x \|p(x')\|dx'} \quad \text{ (decaying)}$$ |
         
     === "Negative slope"
 
         | $E < V(x)$ | $E > V(x)$ |
         | :-----------: | :-----------: |
-        | $$\text{(blowing up) } \quad \frac{C}{\sqrt{\|p(x)\|}} e^{-\frac{1}{\hbar} \int_{x}^{x_t} \|p(x')\|dx'}$$  | $$\frac{2C}{\sqrt{p(x)}} \sin\left( \frac{1}{\hbar} \int_{x_t}^{x} p(x') dx' + \frac{\pi}{4} \right)$$ |
-        | $$ \text{(decaying) } \quad \frac{D}{\sqrt{\|p(x)\|}} e^{\frac{1}{\hbar} \int_{x}^{x_t} \|p(x')\|dx'}$$ | $$ \frac{D}{\sqrt{p(x)}} \cos\left( \frac{1}{\hbar} \int_{x_t}^{x} p(x') dx' + \frac{\pi}{4} \right)$$ |
+        | $$\text{(decaying) } \quad \frac{C}{\sqrt{\|p(x)\|}} e^{-\frac{1}{\hbar} \int_{x}^{x_t} \|p(x')\|dx'}$$  | $$\frac{2C}{\sqrt{p(x)}} \sin\left( \frac{1}{\hbar} \int_{x_t}^{x} p(x') dx' + \frac{\pi}{4} \right)$$ |
+        | $$ \text{(blowing up) } \quad \frac{D}{\sqrt{\|p(x)\|}} e^{\frac{1}{\hbar} \int_{x}^{x_t} \|p(x')\|dx'}$$ | $$ \frac{D}{\sqrt{p(x)}} \cos\left( \frac{1}{\hbar} \int_{x_t}^{x} p(x') dx' + \frac{\pi}{4} \right)$$ |
 
     **More details in the previous lecture note:** [WBK approximation - Connection formulas](wkb_connection.md#434-summary-of-the-connection-formulas)
 
@@ -125,15 +125,15 @@ Below, we will see that the boundary conditions *(connection formulas)* impose c
     === "Positive slope"
         | $E > V(x)$ | $E < V(x)$ |
         | :-----------: | :-----------: |
-        | $$\frac{2C}{\sqrt{p(x)}} \sin\left( \frac{1}{\hbar} \int_x^{x_t} p(x') dx' + \frac{\pi}{4} \right)$$ | $$\frac{C}{\sqrt{\|p(x)\|}} e^{-\frac{1}{\hbar} \int_{x_t}^x \|p(x')\|dx'} \quad \text{ (decaying)}$$ |
-        | $$\frac{D}{\sqrt{p(x)}} \cos\left( \frac{1}{\hbar} \int_x^{x_t} p(x') dx' + \frac{\pi}{4} \right)$$ | $$\frac{D}{\sqrt{\|p(x)\|}} e^{\frac{1}{\hbar} \int_{x_t}^x \|p(x')\|dx'} \quad \text{ (blowing up)}$$ |
+        | $$\frac{2C}{\sqrt{p(x)}} \sin\left( \frac{1}{\hbar} \int_x^{x_t} p(x') dx' + \frac{\pi}{4} \right)$$ | $$\frac{C}{\sqrt{\|p(x)\|}} e^{-\frac{1}{\hbar} \int_{x_t}^x \|p(x')\|dx'} \quad \text{ (blowing up)}$$ |
+        | $$\frac{D}{\sqrt{p(x)}} \cos\left( \frac{1}{\hbar} \int_x^{x_t} p(x') dx' + \frac{\pi}{4} \right)$$ | $$\frac{D}{\sqrt{\|p(x)\|}} e^{\frac{1}{\hbar} \int_{x_t}^x \|p(x')\|dx'} \quad \text{ (decaying)}$$ |
         
     === "Negative slope"
 
         | $E < V(x)$ | $E > V(x)$ |
         | :-----------: | :-----------: |
-        | $$\text{(blowing up) } \quad \frac{C}{\sqrt{\|p(x)\|}} e^{-\frac{1}{\hbar} \int_{x}^{x_t} \|p(x')\|dx'}$$  | $$\frac{2C}{\sqrt{p(x)}} \sin\left( \frac{1}{\hbar} \int_{x_t}^{x} p(x') dx' + \frac{\pi}{4} \right)$$ |
-        | $$ \text{(decaying) } \quad \frac{D}{\sqrt{\|p(x)\|}} e^{\frac{1}{\hbar} \int_{x}^{x_t} \|p(x')\|dx'}$$ | $$ \frac{D}{\sqrt{p(x)}} \cos\left( \frac{1}{\hbar} \int_{x_t}^{x} p(x') dx' + \frac{\pi}{4} \right)$$ |
+        | $$\text{(decaying) } \quad \frac{C}{\sqrt{\|p(x)\|}} e^{-\frac{1}{\hbar} \int_{x}^{x_t} \|p(x')\|dx'}$$  | $$\frac{2C}{\sqrt{p(x)}} \sin\left( \frac{1}{\hbar} \int_{x_t}^{x} p(x') dx' + \frac{\pi}{4} \right)$$ |
+        | $$ \text{(blowing up) } \quad \frac{D}{\sqrt{\|p(x)\|}} e^{\frac{1}{\hbar} \int_{x}^{x_t} \|p(x')\|dx'}$$ | $$ \frac{D}{\sqrt{p(x)}} \cos\left( \frac{1}{\hbar} \int_{x_t}^{x} p(x') dx' + \frac{\pi}{4} \right)$$ |
 
     **More details in the previous lecture note:** [WBK approximation - Connection formulas](wkb_connection.md#434-summary-of-the-connection-formulas)
 

src/wkb_connection.md

@@ -316,8 +316,8 @@ fig.show()
 
 | $E < V(x)$ | $E > V(x)$ |
 | :-----------: | :-----------: |
-| $$\text{(blowing up) } \quad \frac{C}{\sqrt{\|p(x)\|}} e^{-\frac{1}{\hbar} \int_{x}^{x_t} \|p(x')\|dx'}$$  | $$\frac{2C}{\sqrt{p(x)}} \sin\left( \frac{1}{\hbar} \int_{x_t}^{x} p(x') dx' + \frac{\pi}{4} \right)$$ |
-| $$ \text{(decaying) } \quad \frac{D}{\sqrt{\|p(x)\|}} e^{\frac{1}{\hbar} \int_{x}^{x_t} \|p(x')\|dx'}$$ | $$ \frac{D}{\sqrt{p(x)}} \cos\left( \frac{1}{\hbar} \int_{x_t}^{x} p(x') dx' + \frac{\pi}{4} \right)$$ |
+| $$\text{(decaying) } \quad \frac{C}{\sqrt{\|p(x)\|}} e^{-\frac{1}{\hbar} \int_{x}^{x_t} \|p(x')\|dx'}$$  | $$\frac{2C}{\sqrt{p(x)}} \sin\left( \frac{1}{\hbar} \int_{x_t}^{x} p(x') dx' + \frac{\pi}{4} \right)$$ |
+| $$ \text{(blowing up) } \quad \frac{D}{\sqrt{\|p(x)\|}} e^{\frac{1}{\hbar} \int_{x}^{x_t} \|p(x')\|dx'}$$ | $$ \frac{D}{\sqrt{p(x)}} \cos\left( \frac{1}{\hbar} \int_{x_t}^{x} p(x') dx' + \frac{\pi}{4} \right)$$ |
 
 !!! warning "Warning"
     In these tables, we also indicated which wave functions are decaying and which are blowing up *as you move away from the turning point*.