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The Interpretation of the Wave Function of Schroedinger's Equation as the Square Root of the Time-Spent Probability Density Distribution |
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In the mid-1920's Erwin Schroedinger, with a specialization in optics, read de Broglie's work on the wave nature of particles and set about formulating an equation for such entities. The equation was in terms of a wave function φ which Schroedinger did specify but thought that it would be some electromagnetic quantity corresponding to a similar variable for radiation. Max Born speculated that Schroedinger's |φ|² was a probability density function. Born and Pascual Jordan at Göttingen University worked out the implications of this interpretation of Schroedinger's φ. They communicated their speculation to Neils Bohr and Werner Heisenberg in Copenhagen and Bohr replied that they in Copenhagen never considered φ to be anything else. Subsequently this interpretation of φ became known as the Copenhagen Interpretation. Heisenberg discovered the Uncertainty Principle which led him to argue that in an atome an electron does not have a well defined trajectory and instead exists only as a probability density distribution until some interaction causes it to take a definite position.
Here it is argued that Schroedinger's wave function φ does indeed correspond to a probability density function but it is simply the probability density of finding the electron at each position on its periodic trajectory. This happens to be inversely proportional to the absolute value of the particle's velocity. This probability density function will be called the time-spent probability density function.
The purpose of this webpage is to setup the effort to establish the time-spent probrobability density distribution as a valid interpretation of the solutions of Schroedinger's equations. Establishing something as a valid interpretation is quite bit different than proving a theorem. Certain minimal requirements have to be met.
Consider a single particle with a Hamiltonian function H which executes a periodic trajectory. This Hamiltonian function is the total energy of the system in terms of the particles position R and its momentum P. The solution of the Hamiltonian equations for the system gives the particle's trajectory as a function of time. The particle's velocity can then be converted into the time-spent probability function for particle position, D(R). There is also a corresponding probability density function for particle velocity. These two distributions together satisfy the conditions of the Uncertainty Principle. This is the macroscopic side of the problem and the Hamiltonian equations are classical physics.
On the microscopic or quantum level there is the time independent Schroedinger equation for the system. Its solution gives the Schroedinger wave functions whose absolute value squared is a probability density function. Its spatially averaged value corresponds to the time-spent probabilty density function.
To illustrate what is involved consider a harmonic oscillator. Let X be the position of the particle and P the momentum of its mass m. It Hamiltonian function is
where k is the spring constant for the oscillator. When this H function is converted into an operator and the corresponding Schroedinger equation solved the probability density function looks like the following.
The thin line is the probability density function |φ|² from the solution to the time independent Schroedinger equation. The thick line is the time-spent probabilty density function based on the classical harmonic oscillator of the same energy. Thus it is seen that a spatial average of the quantum probability density function corresponds to the classical probability density function. More details need to be considered but now the overall relationships can be displayed as in the following table.
The Relationships between the Dynamics of a Particle on a Micro (quantum) Scale and a Macro (classical) Scale |
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Level | Equations of Trajectory | Connection | Probability Density Distribution |
Macro | Hamiltonian Equations (dR/dt)=∂H/∂P (dP/dt)=−∂H/∂R |
D=1/(T|V|) | D(R) |
Connection | R(t)=∫w(τ)r(t-τ)dτ | operator J equal to H with P→( | V(R) = 1/(T∫w(ρ)|φ|²(R-ρ)dρ) |
Micro | Unknown | |φ|²=1/(T|v|) | Schroedinger's Time Independent Equation Jφ=Eφ |
One detail that needs to be considered is that the correspondence of the spatial average of the quantum probability to the classical depends upon the level of energy of the harmonic oscillator. The previous graph was for 60 quantum units of energy and the correspeondence is quite close. Shown below is the case for 4 quantum units and the corrspondence is much less particularly at the end points of the distributions.
So the proposition that a spatial avarage of the quantum probabiility density distribution being equal to the classical probability density distribution is strictly true only in the limit as the energy of the system increases without bound. This matter is covered in the following studies.
To illustrate the character of the quantum level trajectory of a particle in contrast to the character of the macro level trajectory consider the following graph.
In this graph the nearly flat portion represent what are called allowable quantum states; the steep portions of the trajectory represent what the Copenhagen Interpretation calls quantum jumps. They are not instantaneous transition between allowed quantum states but instead relatively rapid transitions. For an appropriate weighting function w(τ) the time averaged line would be strictly a straight line.
Where σ_{x} and σ_{p} are the standard deviations of the distributions of x and p, respectively, and h is Planck's constant.
(To be continued.)
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