Time-Independent Perturbation Theory (Non-Degenerate) #

This content is primarily based on Section 8, “Approximation Method,” in Chapter 2 of Physics of Atoms and Molecules (2nd Edition) by B. H. Bransden and C. J. Joachain¹.

Physics of Atoms and Molecules

B. H. Bransden and C. J. Joachain, Benjamin Cummings

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1. Summary #

We aim to solve the time-independent Schrödinger equation (TISE):

$$ \begin{align} (\hat{H}_0+\hat{H}’)\varphi = E \varphi \end{align} $$

to determine the $k$-th eigenenergy $E_k$ and eigenfunction $\varphi_k$.

The effect of $\hat{H}’$ on $\hat{H}_0$ is expressed in terms of the order parameter $\lambda \ll 1$, where $\hat{H}_0 = O(\lambda^0)$ and $\hat{H}’ = O(\lambda^1)$.

The solution $E_k$ and $\varphi_k$ of the TISE \eqref{e20} are expanded as a power series in $\lambda$:

$$ \begin{align} \varphi_k &= \varphi_k^{(0)} + \varphi_k^{(1)} + \varphi_k^{(2)} + \cdots \\ E_k &= E_k^{(0)} + E_k^{(1)} + E_k^{(2)} + \cdots \end{align} $$

Here, $E^{(0)}$ and $\varphi^{(0)}$ are solutions of $\hat{H}_0\varphi^{(0)} = E^{(0)} \varphi^{(0)}$, and $E_k^{(s)} = O(\lambda^s)$, $\varphi_k^{(s)} = O(\lambda^s)$ denote terms of order $\lambda^s$. The solution can then be expressed as:

$$ \begin{align} \varphi_k^{(s)}&=\sum_{m\ne k} a_{mk}^{(s)}\varphi_m^{(0)} \\ E_k^{(s)}&=\langle\varphi_k^{(0)}|\hat{H}’|\varphi_k^{(s-1)}\rangle = \sum_{m\ne k} H’_{km}a_{mk}^{(s-1)} \end{align} $$

where the following substitutions are made:

$$ \begin{align} a_{mk}^{(s)} =\frac{1}{E_m^{(0)}-E_k^{(0)}}\biggl[\sum_{j=1}^{s-1} E_k^{(s-j)}a_{mk}^{(j)} - \sum_{n} H’_{mn} a_{nk}^{(s-1)} \biggr],~~(s\geq 1, m\ne k) \end{align} $$

$$ \begin{align} H’_{lm} = \langle\varphi_l^{(0)}|\hat{H}’|\varphi_m^{(0)}\rangle \end{align} $$

For specific problems, truncating the perturbative solution at an appropriate order results in the wavefunction evolving as follows.

2. Problem Setting #

We consider a system described by a time-independent Hamiltonian:

$$ \begin{align} \label{e1} \hat{H}=\hat{H}_0+\hat{H}’ \end{align} $$

and aim to solve the time-independent Schrödinger equation (TISE) to determine the eigenenergy $E = E_k$ and the corresponding eigenfunction $\varphi = \varphi_k$ of the system.

In other words, the TISE we aim to solve in this study is:

$$ \begin{align} \label{e2} \hat{H}\varphi = E \varphi \end{align} $$

The system Hamiltonian \eqref{e1} is expressed as the sum of two Hamiltonians, distinguished and named as follows:

• $\hat{H}_0$: The unperturbed Hamiltonian, which is of order $\lambda^0$.
• $\hat{H}’$: The perturbation Hamiltonian, which is of order $\lambda^1$.

The difference between $\hat{H}_0$ and $\hat{H}’$ lies in the degree of influence $\lambda$ that they exert on the total system Hamiltonian $\hat{H}$.

Here, $\lambda$ is a small parameter, $\lambda \ll 1$, representing a relative magnitude of $\hat{H}’$ to $\hat{H}_0$. In the strict limit $\lambda = 0$, the system reduces to $\hat{H} = \hat{H}_0$, and the perturbation Hamiltonian $\hat{H}’$ vanishes.

Furthermore, we assume that the TISE for the unperturbed Hamiltonian $\hat{H}_0$ has already been solved. That is:

$$ \begin{align} \label{e3} \hat{H}_0 \varphi_k^{(0)} = E_k^{(0)} \varphi_k^{(0)} \end{align} $$

The eigenenergies $E_k^{(0)}$ and the corresponding eigenfunctions $\varphi_k^{(0)}$ are assumed to be known. The operator $\hat{H}_0$ is assumed to be Hermitian, ensuring that the eigenenergies are real.

The eigenfunctions $\varphi_k^{(0)}$ are normalized and orthogonal, satisfying:

$$ \begin{align} \label{e4} \langle\varphi_n^{(0)}|\varphi_m^{(0)}\rangle &= \delta_{n,m},&&(\text{For discrete state}) \\ &= \delta(n-m),&&(\text{For continuum state}) \end{align} $$

For discrete states, $\delta_{i,j}$ refers to the Kronecker delta, while for continuous states, it represents the Dirac delta function. In subsequent discussions, orthogonality will be expressed using $\delta_{i,j}$, which implicitly includes the continuous case.

Under these conditions, we proceed to solve TISE \eqref{e2}. While the approach differs significantly for degenerate and non-degenerate cases, this study focuses solely on the non-degenerate case.

2.1. Solution Method #

The problem to solve is restated as follows:

$$ \begin{align} \label{e5} \left(\hat{H}_0+\hat{H}’\right)\varphi = E \varphi \end{align} $$

Considering the limit $\lambda \to 0$, the $k$-th eigenstate $\varphi_k$ of TISE \eqref{e5} is assumed to deviate only slightly from $\varphi_k^{(0)}$. This forms the basis for perturbation theory. Conversely, perturbation theory should not be applied to problems where $\varphi_k^{(0)}$ is significantly distorted².

In the vicinity of $\lambda \to 0$, the solutions of TISE \eqref{e5} can be expanded as a power series in $\lambda$:

$$ \begin{align} E_k &= E_k^{(0)} + E_k^{(1)} + E_k^{(2)} + \cdots \label{e6}\\ \varphi_k &= \varphi_k^{(0)} + \varphi_k^{(1)} + \varphi_k^{(2)} + \cdots \label{e7} \end{align} $$

where $E_k^{(s)}$ and $\varphi_k^{(s)}$ are terms of order $\lambda^s$. Since perturbation theory assumes $\lambda \ll 1$, the contributions of these terms diminish with increasing $s$, as indicated by $\lambda^s$.

Substituting the expansions \eqref{e6} and \eqref{e7} into TISE \eqref{e5} for the $k$-th eigenenergy $E = E_k$ and eigenfunction $\varphi = \varphi_k$, we obtain:

$$ \begin{align} \label{e8} \left(\hat{H}_0+\hat{H}’\right)\left(\varphi_k^{(0)} + \varphi_k^{(1)} +\cdots\right) = \left(E_k^{(0)} + E_k^{(1)} + \cdots\right) \left(\varphi_k^{(0)} + \varphi_k^{(1)} +\cdots\right) \end{align} $$

By noting that $O(\lambda^s)O(\lambda^r) = O(\lambda^{s+r})$, we match terms of the same order in $\lambda$ on both sides to derive identities:

$$ \begin{align} \lambda^{0}&: &&\hat{H}_0\varphi_k^{(0)} = E_k^{(0)}\varphi_k^{(0)} \label{e9_0} \\ \lambda^{1}&: &&\hat{H}_0\varphi_k^{(1)}+\hat{H}’\varphi_k^{(0)} = E_k^{(0)}\varphi_k^{(1)} + E_k^{(1)}\varphi_k^{(0)} \label{e9_1} \\ \lambda^{2}&: &&\hat{H}_0\varphi_k^{(2)}+\hat{H}’\varphi_k^{(1)} = E_k^{(0)}\varphi_k^{(2)} + E_k^{(1)}\varphi_k^{(1)} + E_k^{(2)}\varphi_k^{(0)} \label{e9_2} \\ & &&\vdots \nonumber \\ \lambda^{s}&: &&\hat{H}_0\varphi_k^{(s)}+\hat{H}’\varphi_k^{(s-1)} = \sum_{j=0}^s E_k^{(s-j)}\varphi_k^{(j)} \label{e9_s}\\ & &&\vdots \nonumber \end{align} $$

For a general consideration, we examine \eqref{e9_s}. Multiplying both sides of \eqref{e9_s} from the left by $\psi_l^{(0)*}$ and integrating over the entire space, we derive:

$$ \begin{gather} \langle\varphi_l^{(0)}|\hat{H}_0|\varphi_k^{(s)}\rangle +\langle\varphi_l^{(0)}|\hat{H}’|\varphi_k^{(s-1)}\rangle =\sum_{j=0}^s E_k^{(s-j)}\langle\varphi_l^{(0)}|\varphi_k^{(j)}\rangle \label{e10} \end{gather} $$

and hence:

$$ \begin{align} &\left(E_l^{(0)}-E_k^{(0)}\right)\langle\varphi_l^{(0)}|\varphi_k^{(s)}\rangle -E_k^{(s)}\delta_{l,k} \nonumber\\ &\hspace{10em}=-\langle\varphi_l^{(0)}|\hat{H}’|\varphi_k^{(s-1)}\rangle +\sum_{j=1}^{s-1} E_k^{(s-j)}\langle\varphi_l^{(0)}|\varphi_k^{(j)}\rangle \label{e11} \end{align} $$

For the case $l = k$, the first term on the left-hand side of \eqref{e11} vanishes, requiring a separate treatment for $l = k$ and $l \ne k$.

2.1.1. Eigenenergy Corrections #

Consider the case $l = k$ in \eqref{e11}. Here, among the variables in the equation, only $E_k^{(s)}$ has the highest order $s$, while the others are corrections of orders lower than $s$. This allows us to solve iteratively from $s = 1$ to determine $E_k^{(s)}$ for any $s$.

Specifically, substituting $l = k$ and solving for $E_k^{(s)}$, we obtain:

$$ \begin{align} E_k^{(s)}=\langle\varphi_k^{(0)}|\hat{H}’|\varphi_k^{(s-1)}\rangle -\sum_{j=1}^{s-1} E_k^{(s-j)}\langle\varphi_k^{(0)}|\varphi_k^{(j)}\rangle \label{e12} \end{align} $$

By choosing the wavefunction to satisfy \eqref{e13a}, the second term in \eqref{e12} vanishes, and we can express:

$$ \begin{align} E_k^{(s)}=\langle\varphi_k^{(0)}|\hat{H}’|\varphi_k^{(s-1)}\rangle \label{e12a} \end{align} $$

2.1.2. Eigenfunction Corrections #

Consider the case $l \ne k$ in \eqref{e11}.

In this case, only $\varphi_k^{(s)}$ has the highest order $s$, while the other variables consist of correction terms of orders lower than $s$. Similar to the eigenenergy corrections, we can solve iteratively from $s = 1$.

Rearranging terms, we obtain:

$$ \begin{align} \langle\varphi_l^{(0)}|\varphi_k^{(s)}\rangle =\frac{1}{E_l^{(0)}-E_k^{(0)}}\biggl[-\langle\varphi_l^{(0)}|\hat{H}’|\varphi_k^{(s-1)}\rangle +\sum_{j=1}^{s-1} E_k^{(s-j)}\langle\varphi_l^{(0)}|\varphi_k^{(j)}\rangle\biggr] \label{e13} \end{align} $$

To derive a more explicit result, we expand $\varphi_k^{(s)}$.

Assuming $\varphi_k^{(0)}$ forms an orthonormal basis, we expand $\varphi_k^{(s)}$ as:

$$ \begin{align} \label{e13a} \varphi_k^{(s)}=\sum_{m\ne k} a_{mk}^{(s)}\varphi_m^{(0)},~~(s\geq 1) \end{align} $$

where the goal is to determine the unknown coefficients $a_{mk}^{(s)}$ for $s \geq 1$ and $m \ne k$. The term $m = k$ is omitted in this expansion, which is justified and explained later.

For the state $k$, we have $a_{mk}^{(0)} = \delta_{mk}$.

In terms of orders, $\varphi_m^{(0)}$ is $O(\lambda^0)$, $\varphi_k^{(s)}$ is $O(\lambda^s)$, and thus $a_{mk}^{(s)}$ is also $O(\lambda^s)$.

Substituting \eqref{e13a} into \eqref{e13}, and noting that $a_{lk}^{(s)} = \langle \varphi_l^{(0)} | \varphi_k^{(s)} \rangle$, we obtain:

$$ \begin{align} a_{lk}^{(s)} =\frac{1}{E_l^{(0)}-E_k^{(0)}}\biggl[\sum_{j=1}^{s-1} E_k^{(s-j)}a_{lk}^{(j)} - \sum_{m\ne k} H’_{lm} a_{mk}^{(s-1)} \biggr],~~(s\geq 1, l\ne k) \label{e13d} \end{align} $$

where we define:

$$ \begin{align} \label{e13e} H’_{lm}\equiv \langle\varphi_l^{(0)}|\hat{H}’|\varphi_m^{(0)}\rangle \end{align} $$

Using \eqref{e13a}, the eigenenergy can be rewritten as:

$$ \begin{align} E_k^{(s)} &= \langle\varphi_k^{(0)}|\hat{H}’|\varphi_k^{(s-1)}\rangle -\sum_{j=1}^{s-1} E_k^{(s-j)}\langle\varphi_k^{(0)}|\varphi_k^{(j)}\rangle \label{e13f} \\ &= \sum_{m\ne k} H’_{km}a_{mk}^{(s-1)}\label{e13g} \end{align} $$

During the transformation to \eqref{e13g}, the second term in \eqref{e13f} is eliminated using \eqref{e13a}.

2.1.3. Reason for Excluding the Term $m = k$ in the Expansion of Perturbative Coefficients #

Here, we explain why the term $m = k$ can be excluded in the expansion \eqref{e13a} of the perturbative terms.

If we consider each perturbative term being expanded in the basis functions ${\varphi_m^{(0)}}$, the wavefunction can be rewritten as:

$$ \begin{align} \varphi_k & = \varphi_k^{(0)} + \varphi_k^{(1)} + \cdots \nonumber\\ & = \varphi_k^{(0)} + \sum_{m} a_{mk}^{(1)}\varphi_m^{(0)} + \sum_{m} a_{mk}^{(2)}\varphi_m^{(0)} + \cdots \nonumber\\ & = \sum_{m} d_{mk}\varphi_m^{(0)} \nonumber\\ & = d_{kk}\left(\varphi_k^{(0)} + \sum_{m\ne k} c_{mk}\varphi_m^{(0)}\right) \label{e14b} \end{align} $$

where we set:

$$ \begin{gather} d_{mk}\equiv \sum_{s=1} (\delta_{mk} + a_{mk}^{(s)}),~~~~c_{mk}\equiv\frac{d_{mk}}{d_{kk}} \end{gather} $$

Since any complex multiple of the wavefunction also satisfies the TISE, the factor $d_{kk}$ in \eqref{e14b} can be disregarded, leading to the solution:

$$ \begin{align} \label{e14c} \varphi_k = \varphi_k^{(0)} + \sum_{m\ne k} c_{mk}\varphi_m^{(0)} \end{align} $$

Thus, it is valid to always find the perturbative term $\varphi_k^{(s)}$ such that its component along $\varphi_k^{(0)}$ is zero.

However, the probability density of the wavefunction $\varphi_k$ found using \eqref{e14c} is not normalized to 1. Therefore, after computing the perturbative terms to the required order, it is necessary to renormalize the wavefunction to 1 as needed.

2.2. Summary #

We aim to solve the time-independent Schrödinger equation (TISE):

$$ \begin{align} \label{e20} (\hat{H}_0+\hat{H}’)\varphi = E \varphi \end{align} $$

to determine the $k$-th eigenenergy $E_k$ and eigenfunction $\varphi_k$, considering the case where the effect of $\hat{H}’$ is small compared to $\hat{H}_0$.

The degree of influence of $\hat{H}’$ on $\hat{H}_0$ is expressed in terms of the order parameter $\lambda \ll 1$. In terms of order, we have $\hat{H}_0 = O(\lambda^0)$ and $\hat{H}’ = O(\lambda^1)$.

Given that the TISE for $\hat{H}_0$:

$$ \begin{align} \label{e21} \hat{H}_0\varphi^{(0)} = E^{(0)} \varphi^{(0)} \end{align} $$

yields known eigenenergies $E_k^{(0)}$ and eigenfunctions $\varphi_k^{(0)}$, the solutions $E_k$ and $\varphi_k$ of TISE \eqref{e20} can be expanded as a power series in $\lambda$:

$$ \begin{align} \varphi_k &= \varphi_k^{(0)} + \varphi_k^{(1)} + \varphi_k^{(2)} + \cdots \label{e22}\\ E_k &= E_k^{(0)} + E_k^{(1)} + E_k^{(2)} + \cdots \label{e23} \end{align} $$

where $E_k^{(s)} = O(\lambda^s)$ and $\varphi_k^{(s)} = O(\lambda^s)$. The solutions are then obtained as:

$$ \begin{align} \varphi_k^{(s)}&=\sum_{m\ne k} a_{mk}^{(s)}\varphi_m^{(0)} \label{e24}\\ E_k^{(s)}&=\langle\varphi_k^{(0)}|\hat{H}’|\varphi_k^{(s-1)}\rangle = \sum_{m\ne k} H’_{km}a_{mk}^{(s-1)} \label{e25} \end{align} $$

Here, we define:

$$ \begin{align} \label{e27} a_{mk}^{(s)} =\frac{1}{E_m^{(0)}-E_k^{(0)}}\biggl[\sum_{j=1}^{s-1} E_k^{(s-j)}a_{mk}^{(j)} - \sum_{n} H’_{mn} a_{nk}^{(s-1)} \biggr],~~(s\geq 1, m\ne k) \end{align} $$

$$ \begin{align} \label{e28} H’_{lm} = \langle\varphi_l^{(0)}|\hat{H}’|\varphi_m^{(0)}\rangle \end{align} $$

Below, we explicitly write out several perturbative terms.

2.2.1. First-Order Perturbation Term #

$$ \begin{align} E_k^{(1)}&= H’_{kk} \label{e30}\\ \varphi_k^{(1)}&=\sum_{m\ne k} a_{mk}^{(1)}\varphi_m^{(0)} \label{e31}\\ &a_{mk}^{(1)}=\frac{- H’_{mk}}{E_m^{(0)}-E_k^{(0)}},\hspace{1em}(m\ne k) \label{e32} \end{align} $$

Here, $a_{kk}^{(1)} = 0$.

2.2.2. Second-Order Perturbation Term #

$$ \begin{align} E_k^{(2)}&=\langle\varphi_k^{(0)}|\hat{H}’|\varphi_k^{(1)}\rangle = \sum_{m\ne k} \frac{- |H’_{mk}|^2}{E_m^{(0)}-E_k^{(0)}} \label{e33}\\ \varphi_k^{(2)}&=\sum_{m\ne k} a_{mk}^{(2)}\varphi_m^{(0)} \label{e34} \end{align} $$

$$ \begin{align} a_{mk}^{(2)}&=\frac{1}{E_m^{(0)}-E_k^{(0)}}\biggl[E_k^{(1)}a_{mk}^{(1)} - \sum_{n\ne k} H’_{mn} a_{nk}^{(1)} \biggr] \label{e35}\\ &=\frac{1}{E_m^{(0)}-E_k^{(0)}}\biggl[E_k^{(1)}\frac{- H’_{mk}}{E_m^{(0)}-E_k^{(0)}} + \sum_{n\ne k} \frac{ H’_{mn}H’_{nk}}{E_n^{(0)}-E_k^{(0)}} \biggr],\hspace{1em}(m\ne k) \label{e36} \end{align} $$

Here, $a_{kk}^{(1)} = a_{kk}^{(2)} = 0$ (for all $k$).

2.2.3. s-th Order Perturbation Term #

$$ \begin{align} E_k^{(s)}&=\langle\varphi_k^{(0)}|\hat{H}’|\varphi_k^{(s-1)}\rangle = \sum_{m\ne k} H’_{km}a_{mk}^{(s-1)} \label{e37}\\ \varphi_k^{(s)}&=\sum_{m\ne k} a_{mk}^{(s)}\varphi_m^{(0)} \label{e38} \end{align} $$

$$ \begin{align} \label{e39} a_{mk}^{(s)} =\frac{1}{E_m^{(0)}-E_k^{(0)}}\biggl[\sum_{j=1}^{s-1} E_k^{(s-j)}a_{mk}^{(j)} - \sum_{n\ne k} H’_{mn} a_{nk}^{(s-1)} \biggr],~~(s\geq 1, m\ne k) \end{align} $$

Here, $a_{kk}^{(s)} = 0$ (for $s \geq 1$ and all $k$).

3. Examples #

3.1. Harmonic Oscillator with Perturbation #

To verify the validity of perturbation theory, we examine a harmonic oscillator with an added perturbation.

3.1.1. Analytical Solution #

$$ \begin{align} \label{e40} (\hat{H}_0+\hat{H}’)\varphi = E \varphi \end{align} $$

Here, the unperturbed Hamiltonian is written as:

$$ \begin{align} \label{e41} \hat{H}_0=-\frac{1}{2}\frac{d^2}{dx^2}+\frac{1}{2}x^2 \end{align} $$

and the perturbation Hamiltonian is:

$$ \begin{align} \hat{H}’&=-ax+\frac{1}{2}a^2+b \label{e42}\\ &\equiv Ax+B \label{e43} \end{align} $$

where $A = -a$ and $B = \frac{1}{2}a^2 + b$ are introduced for computational convenience.

Without considering perturbations, the Schrödinger equation is transformed as:

$$ \begin{gather} \label{e44} \left(-\frac{1}{2}\frac{d^2}{dx^2}+\frac{1}{2}(x-a)^2\right)\varphi(x) = (E-b) \varphi(x) \end{gather} $$

This implies that $a$ shifts the eigenfunctions by $a$, and $b$ shifts the eigenenergies by $b$. The solution to the TISE can be expressed using the unperturbed eigenenergies $E^{(0)}$ and eigenfunctions $\varphi^{(0)}(x)$ as:

$$ \begin{align} \varphi(x)&=\varphi^{(0)}(x-a) \label{e45} \\ E&=E^{(0)}+b \label{e46}\\ &=E^{(0)}+B-\frac{A^2}{2} \end{align} $$

We will now verify whether the results derived using perturbation theory match \eqref{e45} and \eqref{e46}.

The eigenenergies and eigenstates of the TISE $\hat{H}_0\varphi^{(0)} = E^{(0)}\varphi^{(0)}$ are given explicitly as:

$$ \begin{gather} E^{(0)}_n=n+\frac{1}{2}\\ \varphi^{(0)}_n(x)=\frac{\pi^{-1/4}}{\sqrt{2^n n!}}H_n(x)e^{-x^2/2} \end{gather} $$

3.1.2. Results from Perturbation Theory #

We now calculate the results following perturbation theory. The first-order perturbation results, obtained using \eqref{e30}, \eqref{e31}, and \eqref{e32}, are calculated as follows.

The calculation of $H’_{lm}$ yields below.Details of the calculations are provided in the appendix, specifically in equation \eqref{x61}:

$$ \begin{align} H’_{lm} &= \langle\varphi_l^{(0)}|(Ax+B)|\varphi_m^{(0)}\rangle \nonumber\\ &= A\sqrt{\frac{m}{2}}\delta_{l,m-1} + B\delta_{l,m} + A\sqrt{\frac{m+1}{2}}\delta_{l,m+1} \label{e61} \end{align} $$

Additionally, $H’{l,m} = H’{m,l}$ holds.

From these results, the corrections to the eigenenergies and eigenfunctions derived via perturbation theory are given as follows:

• First-Order Perturbation

  • Correction to Eigenenergy

    $$ \begin{align} E_k^{(1)}= H’_{kk} = B \label{e62} \end{align} $$

  • Correction to Eigenfunction

    $$ \begin{align} \varphi_k^{(1)}(x)&=\frac{A}{\sqrt{2}}\left[\sqrt{k}\varphi_{k-1}^{(0)}(x) - \sqrt{k+1} \varphi_{k+1}^{(0)}(x)\right] \label{e63} \\ &=-a \frac{d\varphi_k^{(0)}}{dx} \end{align} $$

    *Note: Utilizes $A = -a$ and the recurrence relation for $\varphi_{k}^{(0)}(x)$ (Appendix \eqref{x80}).*

  • Coefficients Related to the Correction of Eigenfunctions

    $$ \begin{align} a_{mk}^{(1)} = A\sqrt{\frac{k}{2}}\delta_{m,k-1} - A\sqrt{\frac{k+1}{2}}\delta_{m,k+1} \label{e64} \end{align} $$

    *Note: The transformation steps for $a_{mk}^{(1)}$ are detailed in Appendix \eqref{x64}.*

  • Wavefunction Including First-Order Perturbation

    Based on the above, the wavefunction up to first-order perturbation is expressed as:

    $$ \begin{align} \varphi_k(x)&=\varphi_k^{(0)}(x)+\varphi_k^{(1)}(x) + O(\lambda^2)\\ &= \varphi_k^{(0)}(x) + \frac{A}{\sqrt{2}}\left[\sqrt{k}\varphi_{k-1}^{(0)}(x) - \sqrt{k+1} \varphi_{k+1}^{(0)}(x)\right] \\ &=\varphi_k^{(0)}(x)- a \frac{d\varphi_k^{(0)}}{dx} + O(\lambda^2) \label{e80} \\ \end{align} $$

    This result matches the Taylor expansion of the analytical solution \eqref{e45}:

    $$ \begin{align} \varphi^{(0)}(x-a) &= \varphi^{(0)}(x) - a\frac{d\varphi^{(0)}}{dx} + a^2\frac{1}{2!}\frac{d^2\varphi^{(0)}}{dx^2} + O(\lambda^3) \end{align} $$

    up to the $O(\lambda^1)$ term.

    If the probability density is normalized to $1$, the wavefunction becomes:

    $$ \begin{align} \small \varphi_k(x)=\frac{1}{\sqrt{1+\frac{A^2}{2}(2k+1)}}\left(\varphi_k^{(0)}(x) + \frac{A}{\sqrt{2}}\left[\sqrt{k}\varphi_{k-1}^{(0)}(x) - \sqrt{k+1} \varphi_{k+1}^{(0)}(x)\right]\right) + O(\lambda^2) \end{align} $$

• Second-Order Perturbation

  • Correction to Eigenenergy

    $$ \begin{align} E_k^{(2)}&=\langle\varphi_k^{(0)}|\hat{H}’|\varphi_k^{(1)}\rangle =\frac{A}{\sqrt{2}}\left[\sqrt{k}H’_{k,k-1} -\sqrt{k+1} H’_{k,k+1}\right] \nonumber \\ &=-\frac{A^2}{2} \label{e65} \end{align} $$

  • Correction to Eigenfunction

    $$ \begin{align} \label{e66} \varphi_k^{(2)}=\frac{A^2}{4}\biggl[\sqrt{k(k-1)}\varphi_{k-2}(x) + \sqrt{(k+1)(k+2)}\varphi_{k+2}(x) \biggr] \end{align} $$

  • Coefficients Related to the Correction of Eigenfunctions

    $$ \begin{align} \label{e67} a_{mk}^{(2)} = \frac{1}{2}\frac{A^2}{2}\biggl[\sqrt{k(k-1)}\delta_{m,k-2} + \sqrt{(k+1)(k+2)}\delta_{m,k+2} \biggr] \end{align} $$

    *Note: The transformation steps for $a_{mk}^{(2)}$ are detailed in Appendix \eqref{x67a} to \eqref{x67b}.*

  • Wavefunction Including Second-Order Perturbation

    $$ \begin{align} \varphi_k(x)&=\varphi_k^{(0)}(x)+\varphi_k^{(1)}(x)+\varphi_k^{(2)}(x) + O(\lambda^3)\\ &=\left[1+\frac{a^2}{2}\left(k+\frac{1}{2}\right)\right]\varphi_k^{(0)} - a\frac{d\varphi_k^{(0)}}{dx} + \frac{a^2}{2}\frac{d^2\varphi_k^{(0)}}{dx^2} + O(\lambda^3) \\ &\approx \varphi_k^{(0)} - a\frac{d\varphi_k^{(0)}}{dx} + \frac{a^2}{2}\frac{d^2\varphi_k^{(0)}}{dx^2} + O(\lambda^3) \label{e90}\\ \end{align} $$

    The result in \eqref{e90} matches the Taylor expansion of $\varphi_k^{(0)}(x-a)$ up to the $O(\lambda^2)$ term.

    Regarding the coefficients of $\varphi_k^{(0)}$ in \eqref{e90}, the first term is $O(\lambda^0)$, and the second term is $O(\lambda^2)$, which is considered negligible³.

    If the probability density is normalized to $1$, the wavefunction becomes:

    $$ \begin{align} \varphi_k(x)&=\frac{1}{\sqrt{1+\frac{A^2}{2}(2k+1)+\frac{A^4}{8}(k^2+k+1)}} \nonumber\\ &\times \biggl(\varphi_k^{(0)}(x) + \frac{A}{\sqrt{2}}\left[\sqrt{k}\varphi_{k-1}^{(0)}(x) - \sqrt{k+1} \varphi_{k+1}^{(0)}(x)\right] \nonumber \\ &+ \frac{A^2}{4}\biggl[\sqrt{k(k-1)}\varphi_{k-2}(x) + \sqrt{(k+1)(k+2)}\varphi_{k+2}(x)\biggr]\biggr) + O(\lambda^3) \end{align} $$

• Third-Order Perturbation

  • Correction to Eigenenergy

    $$ \begin{align} E_k^{(3)}&=\langle\varphi_k^{(0)}|\hat{H}’|\varphi_k^{(2)}\rangle = 0 \label{e68} \end{align} $$

    From \eqref{e61}, $H’{l,m}$ has non-zero values only when $l = m$ or $l = m \pm 1$. However, as shown by the results for the second-order correction term \eqref{e66}, only $H’{l,l \pm 2}$ terms can arise, which implies that the value is zero.

  • Correction to Eigenfunction

    $$ \begin{align} \varphi_k^{(3)}&=\frac{A^3}{12\sqrt{2}} \biggl[\sqrt{k(k-1)(k-2)}\varphi_{k-3}(x) + 3(k+1)\sqrt{k}\varphi_{k-1}(x) \nonumber\\ & \hspace{3em} - 3k\sqrt{k+1}\varphi_{k+1}(x) - \sqrt{(k+1)(k+2)(k+3)}\varphi_{k+3}(x)\biggr] \label{e68a} \end{align} $$

    The third-order term of the Taylor expansion of the exact solution is given as:

    $$ \begin{align} -\frac{a^3}{3!}\frac{d^3\varphi_k^{(3)}}{dx^3}&=\frac{A^3}{12\sqrt{2}} \biggl[\sqrt{k(k-1)(k-2)}\varphi_{k-3}(x) - 3k^{3/2}\varphi_{k-1}(x) \nonumber\\ &\hspace{1em} + 3(k+1)^{3/2}\varphi_{k+1}(x) - \sqrt{(k+1)(k+2)(k+3)}\varphi_{k+3}(x)\biggr] \label{e68b} \end{align} $$

    The result \eqref{e68b} was expected to match \eqref{e68a}, but as with the second-order case, there is no agreement. There may be a calculation error, but none has been identified so far.

3.1.3. Visualization #

The wavefunctions truncated at appropriate orders in perturbation theory are visualized as follows:

• For $a=0.5$

• For $a=1.0$

• For $a=2.0$

When incorporating up to the second-order perturbation, the wavefunction for the smaller perturbation magnitude $a = 0.5$ closely matches the analytical solution. However, for the larger perturbation magnitude $a = 2.0$, the approximation significantly deviates from the analytical solution.

4. Appendix #

4.1. Transformation of Equation \eqref{e61} #

$$ \begin{align} \langle\varphi_l^{(0)}|x|\varphi_m^{(0)}\rangle &=\frac{\pi^{-1/2}}{\sqrt{2^{l+m} l!m!}} \int_{-\infty}^{\infty} x H_l(x) H_m(x)e^{-x^2} dx \nonumber\\ &=\frac{\pi^{-1/2}}{\sqrt{2^{l+m} l!m!}} \left[\frac{1}{2}\sqrt{\pi}2^m m!\delta_{l+1,m}+l\sqrt{\pi}2^m m!\delta_{l-1,m}\right] \nonumber\\ &=\sqrt{2^{m-l}}\sqrt{\frac{m!}{l!}}\left(\frac{1}{2}\delta_{l+1,m}+l\delta_{l-1,m}\right) \nonumber\\ &=\sqrt{\frac{m}{2}}\delta_{l,m-1}+\sqrt{\frac{m+1}{2}}\delta_{l,m+1} \label{x61} \end{align} $$

Utilizing the three-term recurrence relation of Hermite polynomials⁴:

$$ \begin{gather} x H_n(x) = \frac{1}{2}H_{n+1}(x) + n H_{n-1}(x),\hspace{1em}(n\geq 0)\label{x62} \\ H_{-1}(x) = 0 \end{gather} $$

and the orthogonality of Hermite polynomials⁵:

$$ \begin{gather} \int H_m(x)H_n(x)e^{-x^2} = \sqrt{\pi}2^n n! \delta_{m,n} \end{gather} $$

4.2. Transformation of Equation \eqref{e64} #

$$ \begin{align} a_{mk}^{(1)} &= \frac{- H’_{mk}}{E_m^{(0)}-E_k^{(0)}}, \hspace{1em} (m\ne k) \nonumber\\ &= \frac{-1}{m-k}\left[A\sqrt{\frac{m}{2}}\delta_{m-1,k} + B\delta_{m,k} + A\sqrt{\frac{m+1}{2}}\delta_{m+1,k}\right] \nonumber\\ &= A\sqrt{\frac{k}{2}}\delta_{m,k-1} - A\sqrt{\frac{k+1}{2}}\delta_{m,k+1} \label{x64} \end{align} $$

4.3. Transformation of Equation \eqref{e80} #

For the orthogonal polynomial $\varphi_k^{(0)}$, the following recurrence relation holds:

$$ \begin{align} \label{x80} \frac{d\varphi_k^{(0)}}{dx} = \sqrt{\frac{k}{2}}\varphi_{k-1}^{(0)}(x) - \sqrt{\frac{k+1}{2}}\varphi_{k+1}^{(0)}(x) \end{align} $$

4.4. Transformation of Equation \eqref{e67} #

To calculate $a_{mk}^{(2)}$, the following sum required for the calculation is evaluated:

$$ \begin{align} \sum_{n\ne k} H’_{mn} a_{nk}^{(1)} &= \sum_{n\ne k}H’_{m,n}\left(A\sqrt{\frac{k}{2}}\delta_{n,k-1}-A\sqrt{\frac{k+1}{2}}\delta_{n,k+1}\right) \nonumber\\ &=H’_{m,k-1}A\sqrt{\frac{k}{2}}-H’_{m,k+1}A\frac{k+1}{2} \nonumber\\ &=\frac{A^2}{2}\sqrt{k(k-1)}\delta_{m,k-2} + AB\sqrt{\frac{k}{2}}\delta_{m,k-1} - \frac{A^2}{2}\delta_{m,k} \nonumber\\ &\hspace{5em} - AB\sqrt{\frac{k+1}{2}}\delta_{m,k+1} - \frac{A^2}{2}\sqrt{(k+1)(k+2)}\delta_{m,k+2}　\label{x67a} \end{align} $$

This results in:

$$ \begin{align} a_{mk}^{(2)} &= \frac{1}{E_m^{(0)}-E_k^{(0)}}\biggl[E_k^{(1)}a_{mk}^{(1)} - \sum_{n\ne k} H’_{mn} a_{nk}^{(1)} \biggr] \nonumber\\ &= \frac{1}{m-k}\biggl[B a_{mk}^{(1)} - \sum_{n\ne k} H’_{mn} a_{nk}^{(1)} \biggr] \nonumber\\ &= \frac{-1}{m-k}\frac{A^2}{2}\biggl[\sqrt{k(k-1)}\delta_{m,k-2}+\delta_{m,k}+\sqrt{(k+1)(k+2)}\delta_{m,k+2}\biggr] \end{align} $$

Specifically, since only the values for $m \ne k$ are needed:

$$ \begin{align} a_{mk}^{(2)} &= \frac{-1}{m-k}\frac{A^2}{2}\biggl[\sqrt{k(k-1)}\delta_{m,k-2}+\sqrt{(k+1)(k+2)}\delta_{m,k+2}\biggr],\hspace{1em}(m\ne k) \nonumber\\ &= \frac{1}{2}\frac{A^2}{2}\biggl[\sqrt{k(k-1)}\delta_{m,k-2} + \sqrt{(k+1)(k+2)}\delta_{m,k+2} \biggr] \label{x67b} \end{align} $$

5. References #