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Gram-Schmidt Orthogonalization Process in Quantum Mechanics 📂Quantum Mechanics

Gram-Schmidt Orthogonalization Process in Quantum Mechanics

Definition

The Gram-Schmidt orthogonalization procedure is a method for creating an orthogonal set from vectors that are not orthogonal to each other.

Formula

Suppose there are two time-independent one-dimensional wave functions u1u_{1}, u2u_{2}. Let’s assume that u1u_{1} and u2u_{2} are normalized and not orthogonal to each other. Then, the following wave function uu is a normalized wave function orthogonal to u1u_{1}.

u=(−∫u1∗u2dx)u1+u21−∣∫u1∗u2dx∣2=−⟨u1∣u2⟩u1+u21−∣⟨u1∣u2⟩∣2 \begin{align*} u &= \dfrac { \displaystyle \left(- \int u_{1}^{\ast} u_{2} dx \right)u_{1} + u_{2}}{\displaystyle \sqrt{ 1-\left| \int u_{1}^{\ast} u_{2} dx \right|^{2}}} \\ &= \dfrac { -\braket{u_{1} | u_{2}} u_{1} + u_{2}}{\sqrt{ 1-\left| \braket{u_{1} | u_{2}} \right|^{2}}} \\ \end{align*}

Explanation

Since the goal is to find a new eigenfunction uu that is orthogonal to u1u_{1} from u1u_{1} and u2u_{2}, let’s denote it first as u=c1u1+c2u2u = c_{1} u_{1} + c_{2}u_{2}. Now, by using the condition that uu is orthogonal to u1u_{1} and the normalization condition of uu, c1c_{1} and c2c_{2} can be obtained.

  • Part 1. uu and u1u_{1} are orthogonal.

    Since uu is orthogonal to u1u_{1}, the following holds true.

    ∫uu1∗dx=∫(c1u1u1∗+c2u2u1∗)dx=∫c1u1u1∗dx+∫c2u2u1∗dx=c1+c2∫u1∗u2dx=0 \begin{align*} \int uu_{1}^{\ast} dx &= \int \left( c_{1} u_{1}u_{1}^{\ast} + c_{2}u_{2}u_{1}^{\ast}\right) dx \\ &= \int c_{1}u_{1} u_{1}^{\ast}dx + \int c_{2}u_{2}u_{1}^{\ast} dx \\ &= c_{1} + c_{2}\int u_{1}^{\ast}u_{2} dx \\ &= 0 \end{align*}

    Thus, we obtain the following.

    c1=−c2∫u1∗u2dx(1) c_{1}=-c_{2} \int u_{1}^{\ast}u_{2} dx \tag{1}

  • Part 2. uu is a normalized function.

    Since uu is a normalized eigenfunction, the following holds true.

    ∫uu∗dx=∫(c1u1+c2u2)(c1u1∗+c2u2∗)dx=(c1)2∫u1u1∗dx+(c2)2∫u2u2∗dx+c1c2(∫u1∗u2dx+∫u1u2∗dx)=(c1)2+(c2)2+c1c2(∫u1∗u2dx+∫u1u2∗dx)=1 \begin{align*} \int uu^{\ast} dx &= \int (c_{1}u_{1}+c_{2}u_{2})(c_{1}u_{1}^{\ast}+c_{2}u_{2}^{\ast})dx \\ &= (c_{1})^{2} \int u_{1}u_{1}^{\ast} dx + (c_{2})^{2} \int u_{2}u_{2}^{\ast} dx + c_{1}c_{2} \left( \int u_{1}^{\ast}u_{2} dx + \int u_{1}u_{2}^{\ast}dx \right) \\ &= (c_{1})^{2} + (c_{2})^{2} + c_{1}c_{2} \left( \int u_{1}^{\ast}u_{2} dx + \int u_{1}u_{2}^{\ast}dx \right) \\ &=1 \end{align*}

    Substituting (1)(1) in the last equation yields the following.

    (c2)2(∫u1∗u2dx)2+(c2)2−(c2)2(∫u1∗u2dx)2−(c2)2∫u1∗u2dx∫u1u2∗dx=1 (c_{2})^{2} \left( \int u_{1}^{\ast} u_{2} dx \right)^{2} + (c_{2})^{2} -(c_{2})^{2} \left( \int u_{1}^{\ast} u_{2} dx\right)^{2} -(c_{2})^{2} \int u_{1}^{\ast} u_{2} dx \int u_{1}u_{2}^{\ast} dx =1

      ⟹  (c2)2−(c2)2∫u1∗u2dx∫u1u2∗dx=1 \implies (c_{2})^{2} -(c_{2})^{2} \int u_{1}^{\ast} u_{2} dx \int u_{1}u_{2}^{\ast} dx =1

      ⟹  (c2)2(1−∣∫u1∗u2dx∣2)=1 \implies (c_{2})^{2} \left( 1- \left| \int u_{1}^{\ast}u_{2} dx \right |^{2} \right)=1

      ⟹  c2=11−∣∫u1∗u2dx∣2 \implies c_{2}=\dfrac{1}{\displaystyle \sqrt{1- \left|\int u_{1}^{\ast}u_{2} dx \right|^{2}}}

    The sign of c2c_{2} chosen to be ++ is simply for convenience; −- could also be chosen. Substituting this into (1)(1) yields c1c_{1}.

    c1=−∫u1∗u2dx1−∣∫u1∗u2dx∣2 c_{1} = \dfrac{\displaystyle -\int u_{1}^{\ast}u_{2} dx }{\displaystyle \sqrt{1- \left|\int u_{1}^{\ast}u_{2} dx \right|^{2}}}

Thus, the function uu, which is orthogonal to u1u_{1} and normalized, is as follows.

u=(−∫u1∗u2dx)u1+u21−∣∫u1∗u2dx∣2 u = \dfrac {\displaystyle \left(- \int u_{1}^{\ast} u_{2} dx \right)u_{1} + u_{2}}{\displaystyle \sqrt{ 1-\left| \int u_{1}^{\ast} u_{2} dx \right|^{2}}}

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