The method of convolutions is used to form smoothed functions from approximate wave functions which are discontinuous or have discontinuous first derivatives. As a parameter ε in the smoothed function decreases, the smoothed function approaches the approximate wave function. The expectation values for physical properties corresponding to the approximate wave function are defined to be the limit as ε approaches zero of the expectation values corresponding to the smoothed function. It is found that if the approximate wave function is discontinuous, the corresponding expectation value for the kinetic energy is infinite. Therefore, it seems unlikely that discontinuous approximate wave functions can ever be useful. However, if the approximate wave function is continuous but has a discontinuity in its first derivative, then, as a result of the discontinuity, there is a contribution δT̄12 to the expectation value of the kinetic energy. For a one-dimensional problem δT̄12=−(ℏ2/2m)ψ*(0) [ψ2′(0)−ψ1′(0)].Here ψ(0) is the value of the approximate wave function at the point zero where the discontinuity in its first derivative occurs, and ψ1′(0) and ψ2′(0) are the first derivatives of ψ taken from the left and from the right, respectively, at this point. Similarly, for an N-dimensional problem having a surface S12=0 over which the first derivatives of the approximate wave function are discontinuous, δT̄12= ∫ surfaceψ*[(∂/∂n) (ψ2−ψ1)]dS12.Here the ∂/∂n is the normal derivative with the normal pointed from region 1 towards region 2.The method of convolutions is used to form smoothed functions from approximate wave functions which are discontinuous or have discontinuous first derivatives. As a parameter ε in the smoothed function decreases, the smoothed function approaches the approximate wave function. The expectation values for physical properties corresponding to the approximate wave function are defined to be the limit as ε approaches zero of the expectation values corresponding to the smoothed function. It is found that if the approximate wave function is discontinuous, the corresponding expectation value for the kinetic energy is infinite. Therefore, it seems unlikely that discontinuous approximate wave functions can ever be useful. However, if the approximate wave function is continuous but has a discontinuity in its first derivative, then, as a result of the discontinuity, there is a contribution δT̄12 to the expectation value of the kinetic energy. For a one-dimensional problem δT̄12=−(ℏ2/2m)ψ*(0) [ψ2′(0)−ψ1′(0)].Here ψ(0) is the value of the approximate wave function at the point zero where the discontinuity in its first derivative occurs, and ψ1′(0) and ψ2′(0) are the first derivatives of ψ taken from the left and from the right, respectively, at this point. Similarly, for an N-dimensional problem having a surface S12=0 over which the first derivatives of the approximate wave function are discontinuous, δT̄12= ∫ surfaceψ*[(∂/∂n) (ψ2−ψ1)]dS12.Here the ∂/∂n is the normal derivative with the normal pointed from region 1 towards region 2.

The Journal of Chemical PhysicsMay 1, 196127 citations

Approximate wave function from approximate non-representable Wigner distributions

Patrick J. Loughlin, Leon W. Cohen

We have recently derived a phase space approximation for wave propagation, and we have shown that this approximation is accurate especially compared to the stationary phase approximation. The method gives an approximate Wigner distribution, which raises the question, is the approximate Wigner distribution representable? If it is, then we can invert it to obtain the corresponding wave function. If it is not, can we obtain an approximate wave function from it, and how good is the approximation? Although the answer to the first question is no, we show that from the Wigner approximation one can recover the exact spatial spectrum magnitude and derivative of the phase. We also consider other methods for obtaining approximate wave functions from the Wigner approximation.We have recently derived a phase space approximation for wave propagation, and we have shown that this approximation is accurate especially compared to the stationary phase approximation. The method gives an approximate Wigner distribution, which raises the question, is the approximate Wigner distribution representable? If it is, then we can invert it to obtain the corresponding wave function. If it is not, can we obtain an approximate wave function from it, and how good is the approximation? Although the answer to the first question is no, we show that from the Wigner approximation one can recover the exact spatial spectrum magnitude and derivative of the phase. We also consider other methods for obtaining approximate wave functions from the Wigner approximation.

Journal of Modern OpticsNov 10, 200813 citations

Wave functions

K. Tamvakis

Cambridge University Press eBooksAug 11, 20050 citations

Wave Functions

WORLD SCIENTIFIC eBooksJul 1, 20060 citations

Best Slater approximation of a fermionic wave function

Alex D. Gottlieb, Norbert J. Mauser, J. M. Zhang

We study the best Slater approximation of an $N$ -fermion wave function analytically. That is, we seek the Slater determinant (constructed out of $N$ orthonormal single-particle orbitals) wave function having largest overlap with a given $N$ -fermion wave function. Some simple lemmas have been established and their usefulness is demonstrated on some structured states, such as the GHZ state. In the simplest nontrivial case of three fermions in six orbitals, which the celebrated Borland-Dennis discovery is about, the best Slater approximation wave function is proven to be built out of the natural orbitals in an interesting way. We also show that the Hadamard inequality is useful for finding the best Slater approximation of some special target wave functions.We study the best Slater approximation of an $N$-fermion wave function analytically. That is, we seek the Slater determinant (constructed out of $N$ orthonormal single-particle orbitals) wave function having largest overlap with a given $N$-fermion wave function. Some simple lemmas have been established and their usefulness is demonstrated on some structured states, such as the GHZ state. In the simplest nontrivial case of three fermions in six orbitals, which the celebrated Borland-Dennis discovery is about, the best Slater approximation wave function is proven to be built out of the natural orbitals in an interesting way. We also show that the Hadamard inequality is useful for finding the best Slater approximation of some special target wave functions.

Oct 19, 20150 citations

We study the best Slater approximation of an

N

-fermion wave function analytically. That is, we seek the Slater determinant (constructed out of

N

orthonormal single-particle orbitals) wave function having largest overlap with a given

N

-fermion wave function. Some simple lemmas have been established and their usefulness is demonstrated on some structured states, such as the GHZ state. In the simplest nontrivial case of three fermions in six orbitals, which the celebrated Borland-Dennis discovery is about, the best Slater approximation wave function is proven to be built out of the natural orbitals in an interesting way. We also show that the Hadamard inequality is useful for finding the best Slater approximation of some special target wave functions.