$n\lambda$ But Not As Neil Bohr

From the post "Not Planck's Constant", we made an analogy of the electron orbit being a spring coil stretched.  If we take this analogy further, where the stretched spring coil results in a smaller helical radius, the total number of turns in the coil is fixed and the total length of the coil is also a constant.

Intuitively, consider the displacement long the helical path to be of two components.  One component $\Delta a_l$ to be along the circumference of the orbit and $\Delta a_t$ to be transverse to the circumference.

$(\Delta a)^2=(\Delta a_l)^2+(\Delta a_t)^2$

So,

$\sum{(\Delta a)^2}=\sum{(\Delta a_l)^2}+\sum{(\Delta a_t)^2}$ --- (*)

where the summation is taken once along the orbit. We know that, when $\Delta a$, $\Delta a_l$ and $\Delta a_t$ are small, and all positive,

$\sum{\Delta a}=L_{or}$

$\sum{\Delta a_l}=L_{cir}$

$\sum{\Delta a_t}=n2\pi r_p$

where $L_{or}$ is the total length of the orbit, $L_{cir}$ is the circumference of the orbit, $n$ is the number of turns along the helical path and $r_p$ is the helical radius.

As,

$\sum{(\Delta a_l)^2}\propto\sum{\Delta a_l}$    which implies,

$\sum{(\Delta a_l)^2}\propto L_{cir}$

and

$\sum{(\Delta a_t)^2}\propto \sum{\Delta a_t}$ which implies,

 $\sum{(\Delta a_t)^2}\propto n2\pi r_p$

When $L_{or}$ is fixed so is $\sum{(\Delta a)^2}$, increasing $L_{cir}$ increases $\sum{\Delta a_l}$ and $\sum{(\Delta a_l)^2}$ that decreases $\sum{(\Delta a_t)^2}$ as the expression (*) shows.  A decrease in $\sum{(\Delta a_t)^2}$ in turns decreases $r_p$ as $n$ is also fixed.  So, an increase in $L_{cir}$ leads to a decrease in $r_p$.

$r_p\propto \cfrac{1}{L_{cir}}$

This can be verified by stretching a helical spring coil with a fixed number of turns.  The small radius of the helix becomes smaller as the length of the coil increases.

Furthermore,

$c=r_p\omega=r_{p}2\pi f =constant$

$r_p\propto \cfrac{1}{f}$

So,

$f\propto L_{cir}$

Frequency increases with increasing orbital circumference/radius.  We also have,

$c=f\lambda_{or}$

where $\lambda_c$ has been redefined as the distance traveled along the circumference of the orbit after a time, $T_p$.

$\lambda_{or}\propto \cfrac{1}{L_{cir}}$

This wavelength decreases with increasing orbital circumference/radius.

The period, $T_p$ was previously taken to be $\cfrac{1}{f}$, the reciprocal of the frequency of the particle wave, $\psi$ formulated in the posts "Maybe Not", "Still Not" and "Particle Wave Duality, Seriously MP".  This is the familiar frequency, $f$ we associate with wave in a straight path,

$f=\cfrac{1}{T_{p}}$   where  $f\lambda=c$

The following diagram shows how a frequency component, $f$ might develop on the passing of a particle in helical motion.  From the post "Double-Bell, Balls",

The constant component, $B_c$,

$B_c=\left( \cfrac { \mu_omc^{ 2 } }{ 6\pi  }  \right) ^{ 1/2 }\cfrac { 1 }{ x^{ 3/2 } }$

where $x$ is measured from the center of the helical circular motion.

$B=\left( \cfrac { \mu _{ o }mc^{ 2 } }{ 6\pi  }  \right) ^{ 1/2 }\left\{ \cfrac { 1 }{ (x+r_{ p }e^{ i2\pi ft })^{ 3/2 } }  \right\}$

$B=\left( \cfrac { \mu _{ o }mc^{ 2 } }{ 6\pi  }  \right) ^{ 1/2 }\left\{ \cfrac { 1 }{ x^{ 3/2 }(1+\cfrac { r_{ p } }{ x } e^{ i2\pi ft })^{ 3/2 } }  \right\} =B_{ c }\cfrac { 1 }{ (1+\cfrac { r_{ p } }{ x } e^{ i2\pi ft })^{ 3/2 } }$

where $\cfrac{r_p}{x}\lt1$ and we use a binomial expansion,

$\cfrac{1}{(1+x)^{3/2}}=1-\cfrac { 3 }{ 2 } x+\cfrac { 15 }{ 8 } x^{ 2 }-\cfrac { 35 }{ 16 } x^{ 3 }+\cfrac { 315 }{ 128 } x^{ 4 }+O(x^5)$

We obtain,

$B=B_c\left\{1-\cfrac { 3 }{ 2 } \left( \cfrac { r_{ p } }{ x } e^{ i2\pi ft } \right) +\cfrac { 15 }{ 8 } \left( \cfrac { r_{ p } }{ x } e^{ i2\pi ft } \right) ^{ 2 }-\cfrac { 35 }{ 16 } \left( \cfrac { r_{ p } }{ x } e^{ i2\pi ft } \right) ^{ 3 }+...\right\}$

Surprisingly, we find that there are other frequency components in the measured frequency,

$f_m=f+2f+3f+...nf+...$

$n=1,2,3,...$

where $f_m$ is the measured frequency in $B$ as the wave passes and $f$, the frequency of the helical circular motion which is also the frequency of the wave.  The harmonic frequencies decrease with an envelop of,

$env=\cfrac { 1 }{ n! } \cfrac {- 3 }{ 2 } (\cfrac { -3 }{ 2 } -1)(\cfrac { -3 }{ 2 } -2)..(\cfrac {- 3 }{ 2 } -(n-1))(\cfrac{r_p}{x})^n$

where $n-1$ the nth harmonic and $n=1$ being the fundamental at $f$.  Given $r_p\lt x$, at high values of $x$ only the fundamental component, $f$ is prominent.

However, in this derivation, the wave bent into a torus need not have an integer multiple of turns or equivalently the length of the orbit be of an integer multiple of wavelength, $\lambda_{or}$.

$2\pi r_{or}\ne n_z\lambda_{or}$

where $n_z=1,2,3...$ and $r_{or}$ is the orbital radius,

At any cross-section of the torus, the phase of the wave can change on each pass.  There is simply no constrain that the phase of the wave along its orbit be fixed; that the wave be a standing wave.  As long as $n$ is a constant, not necessarily an integer we have,

$f\propto{L_{cir}}$

What about the standing wave postulated by Neil Bohr?  What is the constraint necessary?