Special Relativity, Action-Reaction and Free Particle Quantum Mechanics
Fuente:
Zenodo
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Recurso digital |
| Pubblicazione: |
Zenodo
2025
|
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866901252470734848 |
|---|---|
| author | Ruggeri, Francesco R. |
| author_facet | Ruggeri, Francesco R. |
| contents | <p>In (1), we suggested that the special relativistic Lorentz invariant A = -Et+px pointed towards the notion of invariance not simply across frames with different constant velocities, -v, but over all free particles. We thus suggested using dt= hbar/E and dx = hbar/p to demonstrate such an invariance, noting that -Et and px terms are independent (just as (px)x, (py)y and (pz)z terms are for a full p dot r). </p> <p> In this note, we wish to investigate the suggestion of (1) in more detail. First, we note that for a given frame with -v, E and p are related by Ev = p (c=1) (for a particle with rest mass) and so it is difficult to see independence between the two even though they represent different components of a 4-vector. Similarly x’=vt’ for a particle with rest mass and one again does not have independence in trajectory even though one has the 4-vector (x,ct). </p> <p> In order to see independence between E and p, we suggest that one must consider at least two particles with different rest masses, different E values, but the same p, hence yielding the same impulse hit. This, however, suggests two different moving frames, each with a different -v and not one. We ask: How does one introduce two frames in a natural manner? We suggest that if one considers two different rest masses mo1 and mo2 at rest, one has a total p of 0, each particle has equal and opposite momentum, but different energies and there is one frame. Using the notion of action-reaction, one may suggest that if mo1 and mo2 are joined by a spring which is released, they may achieve equal and opposite momenta (linked to two different frames with -v1, -v2) and have different energies. The two have equal momenta, but different E values which, we argue, suggests independence of p and E. This may be verified by noting that the two are associated with independent measurements (impulse hits on a target versus distance traveled in a repulsive potential, as discussed in (1)).</p> <p> Similarly, one may have the same E for two particles, but different momenta. In such a case, a different kind of measurement (distance traveled in a repulsive field) does not distinguish between the E values. We note that the action-reaction approach is directly connected with the concept of conservation of momentum. We argue that this reasoning suggests that E and p are independent even though Ev = p (c=1) for a particle with rest mass might suggest otherwise. As a result, -Edt+p dx, should have dt=hbar/E and dx=hbar/p in order to demonstrate such independence. Otherwise, there is a correlation between the two terms and they are not independent. </p> <p> As a result, we argue that dt=hbar/E and dx=hbar/p, i.e. free particle quantum mechanical properties, are not only linked with the special relativistic Lorentz invariant -Et+px, but also with the notion of conservation of momentum and action-reaction. Even though one may use -Et+px for a single particle, the independence of E and p is really seen through considering at least two particles at rest with different masses which break apart with equal and opposite momenta.</p> <p> </p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_14639116 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Special Relativity, Action-Reaction and Free Particle Quantum Mechanics Ruggeri, Francesco R. <p>In (1), we suggested that the special relativistic Lorentz invariant A = -Et+px pointed towards the notion of invariance not simply across frames with different constant velocities, -v, but over all free particles. We thus suggested using dt= hbar/E and dx = hbar/p to demonstrate such an invariance, noting that -Et and px terms are independent (just as (px)x, (py)y and (pz)z terms are for a full p dot r). </p> <p> In this note, we wish to investigate the suggestion of (1) in more detail. First, we note that for a given frame with -v, E and p are related by Ev = p (c=1) (for a particle with rest mass) and so it is difficult to see independence between the two even though they represent different components of a 4-vector. Similarly x’=vt’ for a particle with rest mass and one again does not have independence in trajectory even though one has the 4-vector (x,ct). </p> <p> In order to see independence between E and p, we suggest that one must consider at least two particles with different rest masses, different E values, but the same p, hence yielding the same impulse hit. This, however, suggests two different moving frames, each with a different -v and not one. We ask: How does one introduce two frames in a natural manner? We suggest that if one considers two different rest masses mo1 and mo2 at rest, one has a total p of 0, each particle has equal and opposite momentum, but different energies and there is one frame. Using the notion of action-reaction, one may suggest that if mo1 and mo2 are joined by a spring which is released, they may achieve equal and opposite momenta (linked to two different frames with -v1, -v2) and have different energies. The two have equal momenta, but different E values which, we argue, suggests independence of p and E. This may be verified by noting that the two are associated with independent measurements (impulse hits on a target versus distance traveled in a repulsive potential, as discussed in (1)).</p> <p> Similarly, one may have the same E for two particles, but different momenta. In such a case, a different kind of measurement (distance traveled in a repulsive field) does not distinguish between the E values. We note that the action-reaction approach is directly connected with the concept of conservation of momentum. We argue that this reasoning suggests that E and p are independent even though Ev = p (c=1) for a particle with rest mass might suggest otherwise. As a result, -Edt+p dx, should have dt=hbar/E and dx=hbar/p in order to demonstrate such independence. Otherwise, there is a correlation between the two terms and they are not independent. </p> <p> As a result, we argue that dt=hbar/E and dx=hbar/p, i.e. free particle quantum mechanical properties, are not only linked with the special relativistic Lorentz invariant -Et+px, but also with the notion of conservation of momentum and action-reaction. Even though one may use -Et+px for a single particle, the independence of E and p is really seen through considering at least two particles at rest with different masses which break apart with equal and opposite momenta.</p> <p> </p> |
| title | Special Relativity, Action-Reaction and Free Particle Quantum Mechanics |
| url | https://doi.org/10.5281/zenodo.14639116 |