Wavefront Shape Generated by Double Slits

Your assumptions are quite insightful and broadly accurate in describing the photon’s behavior in the context of the double-slit experiment, though there are a few subtle refinements to consider:

a. Initial Photon Wavefunction

• When the photon leaves its source, it is best described by a wave packet or spatially extended probability amplitude.
• This wave packet is not a single point; instead, it is a superposition of plane waves with slightly different wavelengths and directions, centered around a dominant wavelength
  $\lambda$ 

  The packet defines the probability density for detecting the photon at any given point.

b. Interaction with the Double Slits

• When the photon encounters the double-slit apparatus, its wavefunction interacts with the geometry of the slits:
  • If the photon’s wavefunction is coherent (i.e., spatially extended enough to “span” both slits), the wavefunction passes through both slits simultaneously as a superposition.
  • The wavefunction at this stage is no longer a simple plane wave; it acquires a spatial pattern determined by the slit geometry (width, separation, etc.).

c. Emanation from the Slits

• After passing through the slits, the wavefunction can be thought of as emerging from two coherent sources, much like radial wavefronts emanating from two point sources.
• This emanation is not strictly circular but depends on:
  1. The width of each slit: Wider slits produce narrower diffraction patterns; narrower slits produce broader diffraction patterns.
  2. The separation between the slits: Determines the spacing of the interference fringes on the detection screen.
  3. The wavelength: Sets the fundamental scale of the interference pattern.

d. Interference on the Screen

• The interference pattern on the screen is determined by the superposition of the two wavefunctions emerging from the slits.
• The pattern is a result of constructive and destructive interference, depending on the relative phase of the wavefunctions from the two slits.

Refinement to Your Assumptions:

1. While the photon can be detected as a point on the screen, its wavefunction is not a point. It is a spatially extended probability amplitude, even when it interacts with the slits.
2. The emanation from the slits is not strictly “radial” but follows the diffraction pattern determined by the slit geometry and wavelength.

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2. Second Question: Does the Schrödinger Wave Equation Predict the Circular Emanation Patterns and Interference?

Yes, the Schrödinger wave equation (SWE) predicts the interference patterns and the emanation characteristics based on the following factors:

a. Geometry of the Slits

The geometry of the slits plays a critical role:

• Slit Width: The width of each slit determines the diffraction envelope. Narrower slits produce broader diffraction patterns, while wider slits produce narrower patterns.
• Slit Separation: The distance between the slits determines the spacing of the interference fringes on the screen.

b. Wavelength

The wavelength \(\lambda\) of the photon sets the fundamental scale of the interference pattern.
The fringe spacing \(\Delta x\) on the detection screen is proportional to \(\lambda\), according to:
$$\Delta x = \frac{\lambda L}{d}$$
Where:
– \(\lambda\): Wavelength of the photon
– (L): Distance from the slits to the detection screen
– (d): Separation between the slits

c. Diffraction and Edge Effects

The SWE inherently accounts for diffraction effects, which occur due to the interaction of the wavefunction with the edges of the slits:

• When the wavefunction encounters a slit, the boundary conditions at the edges of the slit cause the wavefunction to spread out and propagate as a diffraction pattern.
• The diffraction pattern from each slit combines with the other slit’s wavefunction to produce the overall interference pattern.

d. Factors Considered by the SWE

The Schrödinger Wave Equation (SWE) predicts the probability distribution \( \psi(\mathbf{r})^2 \) on the detection screen by solving the wavefunction under the following constraints:

1. Boundary Conditions: The slits impose boundary conditions on the wavefunction. For example, outside the slits, the wavefunction is zero because the photon cannot pass through the opaque material.
2. Superposition Principle: The wavefunction from the two slits interferes constructively or destructively, depending on the path difference and relative phase.
3. Huygens-Fresnel Principle (implicitly): The SWE naturally incorporates the idea that every point on the wavefront emerging from the slits acts as a source of secondary wavelets, leading to the observed diffraction and interference.

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3. Circular Emanation vs. Diffraction Patterns

The emanation patterns from the slits are not purely “circular” but are better described as diffraction patterns. The shape of these patterns depends on:

1. 1. Single-Slit Diffraction:
      • Each slit produces a diffraction pattern described by the Fraunhofer diffraction formula:
        $$I(\theta) \propto \left( \frac{\sin \beta}{\beta} \right)^2$$
        Where: \(\beta = \frac{\pi a \sin \theta}{\lambda}\) and ( a ) is the slit width.
   2. Double-Slit Interference
      • The interference from two slits produces fringes superimposed on the diffraction envelope:

$$I(\theta) \propto \left( \frac{\sin \beta}{\beta} \right)^2 \cos^2\left( \frac{\pi d \sin \theta}{\lambda} \right)$$
Where:

1. 1. • ( d ): Slit separation
      • \(cos^2\left( \frac{\pi d \sin \theta}{\lambda} \right)\): Describes the interference fringes
      • \(\left( \frac{\sin \beta}{\beta} \right)^2\): Describes the single-slit diffraction envelope

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4. Conclusion

First Question:

Your assumptions are broadly accurate, but the photon’s wavefunction is not a point and does not strictly emanate in circular patterns. Instead:

1. 1. • The wavefunction spans the slits and propagates as a superposition of diffracted wavefronts.
      • The interference pattern arises from the superposition of the wavefunctions emerging from the two slits.

Second Question:

The Schrödinger wave equation predicts the interference pattern by accounting for:

1. 1. 1. The slit geometry (width, separation, and spacing).
      2. The wavelength of the photon.
      3. The diffraction effects at the slit edges.

The Schrödinger Wave Equation (SWE) computes the wavefunction \( \psi(\mathbf{r}) \), whose squared magnitude \( \psi(\mathbf{r})^2 \) gives the intensity distribution on the screen. Both the diffraction pattern and interference fringes emerge naturally from the solution of the SWE.