On the exact discretization of Schrödinger equation

In quantum mechanics the time-dependent Schrödinger equation is a second order linear partial differential equation

where the single-particle wavefunction $\Psi(x,t)$ is a function of position $x$ and time $t$, $U(x)$ denotes the potential energy, and $M$ is particle’s mass. Though the Schrodinger equation has a specific form in terms of continuous variables $x$ and $t$, there are different discrete analogues of the equation [1, 2, 3]. For example, by using the standard central difference formula for the Laplace operator

a possible discretization of Eq.(1) is

Here $\Psi(x,t)$ is defined only on the discrete position ${x}=\ell(\hat{1}n_{1}+\hat{2}n_{2}+\hat{3}n_{3})$ ($n_{1},n_{2},n_{3}$ are integers), $\hat{a}$ ($a=1,2,3$) denote the unit base vectors of the Cartesian coordinate system, and $\ell$ is the lattice spacing between the nearest-neighboring spatial sites. Eq.(3) cannot be thought as an exact discrete analog of Eq.(1) since it has different dispersion relation $\varepsilon(k)$ than that in the continuous Schrödinger equation with zero-potential $U(x)=0$ [2]. In other words, the theory described by (3) is different from the theory described by the exact discrete analogue of the Schrödinger equation since both theories do have different Hamiltonian operators in discrete space. They are equal to each other only in the zero-spacing limit $\ell\to 0$.

An exact discretization of Schrodinger equation in one-dimensional space has been derived directly from the continuous Schrodinger equation [3]

Different from the standard central difference formula, the exact discretized Schrödinger equation has difference of integer order that is represented by infinite series, and a long-range interaction is suggested in the discretized equation.

In this paper, we show that a natural way for the derivation of the exact discretized Schrödinger equation is from the Hamiltonian operator of the Schrödinger field theory. It is known that the continuous Schrödinger equation that describes the time evolution of wavefunction can be derived from the following equation

where the Hamiltonian operator $H$ plays the role of the generator of time evolution. The so-called wavefunction $\Psi(x,t)$ is the inner product between the position eigenket $|x\rangle$ and the quantum state $|\Psi\rangle$, $\Psi(x,t)=\langle x|\Psi\rangle$. Once the Hamiltonian operator is given, by taking the inner product on both sides of (5), the Schrödinger equation in (1) is obtained. Following the same line of thought, the exact discretized Schrödinger equation could also be derived from the Hamiltonian operator of the quantum system in discrete space. For example, consider a Schrödinger field theory with the Lagrangian density [4]

the continuous Schrödinger equation is derived as the equation of motion in the theory. Without the self interaction term (ie., $U(x)$=0), after the second quantization of the free theory in discrete space, the free Hamiltonian operator is diagonal in the discrete momentum space

where $\varepsilon_{k}=\hbar^{2}k^{2}/2M$ denotes the energy for a free particle with momentum $\hbar k$, and $a_{k}^{\dagger}$ and $a_{k}$ are the creation and the annihilation operator that satisfy the quantization relation

Here the momentum vector $k$ takes only discrete values. Once the free Hamiltonian operator $H_{0}$ is obtained in momentum representation, it is straightforward to rewrite $H_{0}$ in position representation by using discrete Fourier transforms. Obviously, $H_{0}$ will not be diagonal in position representation due to Heisenberg’s uncertainty relation, thus a spatially localized particle has a chance to hop to other locations instead of staying at the same position in the free theory. With the presence of the self interaction $U(x)$, the Hamiltonian operator becomes

Here the potential energy part of $H$ is diagonal in position representation. Once the Hamiltonian operator $H$ in position representation is obtained, the exact discrete analogue of Schrödinger equation can be derived from (5) by

Consider a discrete position space with a square lattice structure that is described by

Here ${x}$ denotes the position of the spatial lattice sites, $n_{1},n_{2}$, and $n_{3}$ are integers, $\hat{a}$ ($a=1,2,3$) denote the unit base vectors of the Cartesian coordinate system, $\ell$ is the lattice spacing, and $N$ is a large even number. As shown in Fig.1, the three-dimensional discrete position space can be viewed as a cube with side length $L=N\ell$ and total volume $L^{3}$. Given a free Schrödinger field theory in the discrete space with Hamiltonian $H_{0}$, the operator $H_{0}$ is diagonal in momentum space and has the form as described in (7). The diagonal form of $H_{0}$ in momentum space is indeed a direct consequence of the translational symmetry in position space in the free theory. In the free theory, a normalized single particle state with momentum k is

where $|0\rangle$ denotes the vacuum state, and the momentum $k$ takes only discrete values

The integers $m_{1},m_{2}$ and $m_{3}$ range from $-N/2$ to $N/2-1$. The discrete momentum space also has a square lattice structure but with lattice spacing $2\pi/L$.

To transform Hamiltonian from momentum representation into position representation, we define the field operator $a_{x}$ and its hermitian conjugate $a_{x}^{\dagger}$ as the discrete Fourier transforms of $a_{k}$ and $a_{k}^{\dagger}$

Here the discrete momentum ${k}$ is defined in (13), and ${x}$ denotes the position of lattice sites in the three-dimensional discrete position space. From the quantization relation in (8), the field operators $a_{x}$ and $a_{x}^{\dagger}$ are shown to satisfy the commutator relation

The commutator relation in (16) thus allows us to define the position eigenkets as

The position eigenkets are normalized states, like the momentum states $|k\rangle$, can also be single-particle states in the Schrödinger field theory in the discrete space. From (14) and (15), $a_{k}$ and $a_{k}^{\dagger}$ can be written as linear combinations of $a_{x}$ and $a_{x}^{\dagger}$

Taking Eq.s (18) and (19) into $H_{0}$ in (7), we have

The form of $H_{0}$ in (20) can be simplified further if the single-particle energy $\varepsilon_{k}$ is known. In the Schrödinger field theory, $\varepsilon_{k}=\hbar^{2}{k}^{2}/2M$, the sum $(1/N^{3})\sum_{k}\varepsilon_{k}$ in (20) is found to be

Using the fact that

where $k_{a}$ is the $a$-th component of wave vector $k$, $n$ and $n^{\prime}$ are integers. The other sum $(1/2MN^{3})\sum_{k}k^{2}e^{i\vec{k}\cdot(\vec{x}-\vec{x}^{\prime})}$ in (20) with nonzero spatial separation $(\vec{x}-\vec{x}^{\prime})$ is found be nonzero only when $(\vec{x}-\vec{x}^{\prime})$ lies in either direction $\hat{1},\hat{2}$, or $\hat{3}$. Let assume that $\vec{x}-\vec{x}^{\prime}=\hat{1}m\ell$ ($m\neq 0$), then the sum is

In the $N\to\infty$ limit, when $\vec{x}=\vec{x}^{\prime}+\hat{a}m\ell$ ($a=1,2,3$), the sum in (23) becomes

Finally, from Eq.s (20, 21, 24), the free Hamiltonian operator in the large $N$ limit in position representation is

Here $\mathbb{1}=\sum_{x}a_{x}^{\dagger}a_{x}$ denotes the identity operator. $H_{0}$ is non-diagonal in position representation with the presence of the hopping interaction $a^{\dagger}_{x+\hat{a}m\ell}a_{x}$. The hopping interaction terms are responsible for the position-momentum uncertainty relation $\Delta x\Delta p\geq\hbar/2$. With the hopping terms, a spatially localized particle can hop to other places rather than just staying at the same location. Turning on potential energy $U(x)$, the Hamiltonian operator for the Schrödinger field theory is

Similarly, in two dimensional and one dimensional discrete space, the Hamiltonian operators for the corresponding Schrödinger field theories are

The exact discrete analog of Schrodinger equation is derived as follows. Let’s take the one-dimensional Schrödinger theory as an example. Consider a single-particle quantum state $|\Psi\rangle$ in position representation

From (5) and (28), it has

Comparing the coefficient of $|x\rangle$ on both sides of the equation (30), we get

The result in (31) is indeed the exact discrete analogue of Schrödinger equation given in (3). Using the identity

the equation in (31) can be further written as

Eq.(33) shows that the second order differential operation $\partial^{2}\Psi(x,t)/\partial x^{2}$ can be replaced by the infinite difference

in the discrete version of quantum mechanics. In two $(D=2)$ or three $(D=3)$ dimensional space, the exact discretized Schrödinger equation is

Now we know that Eq.(35) is an exact analogue of the continuous Schrödinger equation since it is derived directly from the Hamiltonian operator of the theory in discrete space. The standard central difference equation given in (3) is only an approximate. In fact, Eq.(3) actually describes another quantum theory that has the Hamiltonian operator

with

Obviously, the Hamiltonian operator given in (36, 37, 38) is different from the Hamiltonian operator given in (26). The dispersion relation $\varepsilon_{k}$ in (38)

is effectively the same dispersion relation given in [2]. It differs from the free particle energy $\hbar^{2}k^{2}/2M$ by less than one percent if $|k_{b}\ell|<0.35$. The two Hamiltonian operators indeed describe different quantum theories in discrete space. When $|k_{b}\ell|<0.35$ is satisfied, the theory described by the Hamiltonian in (38) could be a good approximation of the theory that is described by the Hamiltonian in (7). They are equal to each other only in the $\ell\to 0$ limit.

In Schrodinger field theory in discrete space, the position and momentum operators in $\hat{b}$-direction $(b=1,2,3)$ are defined as

With the definition of momentum operator, a position eigenstate $|x_{0}\rangle$ will have a non-zero mean value of $\hat{P}_{b}$

It is due to the asymmetric range of $k_{b}$

The non-zero mean value can be removed either by removing $k_{b}=-\pi/\ell$ from the allowable values of $k_{b}$ or by taking the limit $L\to\infty$. Similarly, a momentum state $|k_{0}\rangle$ also has a non-zero mean $\langle k_{0}|\hat{X}_{b}|k_{0}\rangle=-{\ell}/{2}$ due to the asymmetry of the range of $x_{b}$. It can also be removed either by removing $x_{b}=-L/2$ from the allowable eigenvalues of $\hat{X}_{b}$ or by taking the $\ell\to 0$ limit. In position representation, the momentum operator becomes

The first part of momentum operator in (44)

is also due to the asymmetry of the range of $k_{b}$. It disappears if a symmetric momentum operator is defined

For spatially smooth wavefunction $\Psi(x,t)$ the first part of $\hat{P}_{b}$ has a negligible contribution to $\langle x|\hat{P}_{b}|\Psi\rangle$ since $(-1)^{m}\Psi(x-\hat{b}m\ell,t)$ oscillates extremely rapidly in space as $m$ increases

In the $\sqrt{N}\ell\to\infty$ limit

Here $\Psi(x,t)$ is assumed to be a normalized wavefunction, $\sum_{x}|\Psi(x,t)|^{2}=1$, so that $\sum_{m}|\Psi(x-\hat{b}m\ell,t)|\leq\sqrt{N}$. Therefore, in the $\sqrt{N}\ell\to\infty$ limit for any normalized quantum state $|\Psi\rangle$, the exact discrete analogue of the differential operation becomes [3, 5]

The commutator relation between $\hat{X}_{a}$ and $\hat{P}_{b}$ is found in a similar way. From (40) and (41),

The result in (50) is not zero only when $a=b$. Without loss of generality, let’s consider the case that $a=b=1$. Using

where $k_{1}=2\pi n/L$ and $x_{1}-x^{\prime}_{1}=m\ell\neq 0$, the commutator $[\hat{X}_{1},\hat{P}_{1}]$ is

Here the integer $m$ ranges from $-N/2\leq(x_{1}/\ell-n)\leq N/2-1$. From the above results, we get the commutator relation

The commutator relation shown in (53) differs from the conventional quantization relation $[\hat{X}_{a},\hat{P}_{b}]=i\hbar\delta_{ab}\mathbb{1}$ in continuous space. Since for a Schrödinger theory that is defined on a discrete space with finite number of lattice sites (the total number of sites is $N^{3}$), all momentum eigenstates $|k\rangle$ and position eigenstates $|x\rangle$ are normalized states. This means that all $|k\rangle$ and $|x\rangle$ eigenkets can be used as the single-particle quantum states in the theory if relativistic energy is allowed (For highly localized state $|\Psi\rangle=|x\rangle$, the particle state has energy $\langle\Psi|H|\Psi\rangle=\pi^{2}\hbar^{2}/2M\ell^{2}\gg Mc^{2}$). Thus, the relations that $\langle k|[\hat{X}_{a},\hat{P}_{b}]|k\rangle=0=\langle x|[\hat{X}_{a},\hat{P}_{b}]|x\rangle$ help to explain why the conventional commutator relation $[\hat{X}_{a},\hat{P}_{b}]=i\hbar\delta_{ab}\mathbb{1}$ cannot be correct in discrete space.

When the commutator acts upon a quantum state $|\psi\rangle=\sum_{x}\Psi(x,t)|x\rangle$, it gives

The infinite sum in (54) has extremely small contribution when the normalized wavefunction $\Psi(x,t)$ is a localized and relatively smooth function in position space. In the situation, $(-1)^{m}(\frac{\pi m}{N})(1+i\cot(\frac{\pi m}{N}))\Psi(x+\hat{a}m\ell,t)$ is not only a finite localized function but also oscillates very rapidly in space. So, in the limit that $N\gg 1$ and $\ell\to 0$,

Thus in the continuous limit, for any quantum state $|\Psi\rangle$ with a localized and relatively smooth wavefunction, the conventional commutator relation holds

The problems of a Gaussian wave packet passing through one or two potential barriers have been studied either theoretically or numerically [6, 7]. In the paper a comparison between the exact discrete analogue of Schrödinger equation and the standard central difference equation is made by numerically studying the transmission probability for a Gaussian wave packet passing through a square potential barrier in one dimensional space. We assume the initial wavefunction for a particle to be a Gaussian wave packet with its center located at $x_{0}$

Here $\sigma$ is the standard deviation of particle’s probability distribution at time $t=0$. The initial wave packet $\Psi(x,0)$ actually consists of many plane waves with wave numbers $k$ around $k_{0}$ ($k_{0}>0$)

where $\Phi(k)$ is

Since $\Phi(k)$ is symmetric about $k_{0}$, the average energy for the wave packet is thus $E_{0}=\hbar^{2}k_{0}^{2}/2M$. The wavefunction at a later time is obtained by solving Schrödinger equation. Using the exact discretized Schrödinger equation in (35), the wave function $\Psi(x,t+\Delta t)$ at a later time $t+\Delta t$ can be obtained from the same wavefunction at the previous time $t$