The major problem in quantum physics is to find the wave function of a given physical system and to understand (predict) how the wave function of the system evolves in time, or how it changes under external influences. In 1926, Erwin Schrodinger predicted that the wave function of a given physical system might be completely determined if the total energy of the system was known. Since the equation represents a real physical system, it must satisfy the following conditions: the equation must be linear and coefficients appearing in this equation should only depend on the parameters characteristic of the particle. This chapter focuses on to the non-relativistic case only. It considers the case of a free particle moving in one dimension. The chapter shows that the Schrodinger equation can be obtained from the energy of the free particle. It shows that the Schrodinger equation guarantees the conservation of normalization of the wave function.The major problem in quantum physics is to find the wave function of a given physical system and to understand (predict) how the wave function of the system evolves in time, or how it changes under external influences. In 1926, Erwin Schrodinger predicted that the wave function of a given physical system might be completely determined if the total energy of the system was known. Since the equation represents a real physical system, it must satisfy the following conditions: the equation must be linear and coefficients appearing in this equation should only depend on the parameters characteristic of the particle. This chapter focuses on to the non-relativistic case only. It considers the case of a free particle moving in one dimension. The chapter shows that the Schrodinger equation can be obtained from the energy of the free particle. It shows that the Schrodinger equation guarantees the conservation of normalization of the wave function.

Mar 3, 20170 citations

The conservation laws in quantum mechanics revisited and extended

Richard Joseph Oldani

It is often claimed that Einstein is wrong about quantum mechanics. However, when comparisons are made with respect to theoretical foundations rather than experimental results Einstein’s theories are found to be superior. Although quantum mechanics correctly describes energy exchange due to emissions with remarkable accuracy it ignores momentum exchange (the absorptions), thereby failing to recognize their causal relationship due to the conservation laws. The deficiencies in non-relativistic theory are corrected by introducing Hamilton’s principle and deriving relativistic equations of motion. This allows wave function behavior to be described as the combined action of a particle and an associated localized field. The calculus of variations is applied to the Schrödinger wave equation to show that it is incomplete because it yields twice the allowable action minimum.It is often claimed that Einstein is wrong about quantum mechanics. However, when comparisons are made with respect to theoretical foundations rather than experimental results Einstein’s theories are found to be superior. Although quantum mechanics correctly describes energy exchange due to emissions with remarkable accuracy it ignores momentum exchange (the absorptions), thereby failing to recognize their causal relationship due to the conservation laws. The deficiencies in non-relativistic theory are corrected by introducing Hamilton’s principle and deriving relativistic equations of motion. This allows wave function behavior to be described as the combined action of a particle and an associated localized field. The calculus of variations is applied to the Schrödinger wave equation to show that it is incomplete because it yields twice the allowable action minimum.

QeiosSep 20, 20230 citations

Relativistic nature of the Schrödinger equation

Frank V. Kowalski

Physics Letters ANov 1, 19930 citations

Higher conservation laws for the quantum non-linear Schrödinger equation

Brian Davies

Physica A Statistical Mechanics and its ApplicationsAug 1, 199042 citations

Probability conservation for multi-time integral equations

Matthias Lienert

In relativistic quantum theory, one sometimes considers integral equations for a wave function $ψ(x_1,x_2)$ depending on two space-time points for two particles. A serious issue with such equations is that, typically, the spatial integral over $|ψ|^2$ is not conserved in time -- which conflicts with the basic probabilistic interpretation of quantum theory. However, here it is shown that for a special class of integral equations with retarded interactions along light cones, the global probability integral is, indeed, conserved on all Cauchy surfaces. For another class of integral equations with more general interaction kernels, asymptotic probability conservation from $t=-\infty$ to $t=+\infty$ is shown to hold true. Moreover, a certain local conservation law is deduced from the first result.In relativistic quantum theory, one sometimes considers integral equations for a wave function $ψ(x_1,x_2)$ depending on two space-time points for two particles. A serious issue with such equations is that, typically, the spatial integral over $|ψ|^2$ is not conserved in time -- which conflicts with the basic probabilistic interpretation of quantum theory. However, here it is shown that for a special class of integral equations with retarded interactions along light cones, the global probability integral is, indeed, conserved on all Cauchy surfaces. For another class of integral equations with more general interaction kernels, asymptotic probability conservation from $t=-\infty$ to $t=+\infty$ is shown to hold true. Moreover, a certain local conservation law is deduced from the first result.

arXiv:2210.05759Oct 11, 20220 citations

In relativistic quantum theory, one sometimes considers integral equations for a wave function

ψ(x_1,x_2)

depending on two space-time points for two particles. A serious issue with such equations is that, typically, the spatial integral over

|ψ|^2

is not conserved in time -- which conflicts with the basic probabilistic interpretation of quantum theory. However, here it is shown that for a special class of integral equations with retarded interactions along light cones, the global probability integral is, indeed, conserved on all Cauchy surfaces. For another class of integral equations with more general interaction kernels, asymptotic probability conservation from

t=-\infty

t=+\infty

is shown to hold true. Moreover, a certain local conservation law is deduced from the first result.