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Tom Mattson

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Many moons ago, when I used to teach Modern Physics to engineering undergrads, I told them that to get to quantum mechanics from classical mechanics, we had to start from the Hamiltonian as opposed to Newton's laws. I told them that the reason was that Newton's second law for a particle, namely:

Σ

presupposed that we could know the trajectory

So why not try to quantize Newton?

I want to look at an example first, and then go into generalities later. Let's look at the simple harmonic oscillator in 1D along the x axis.

Writing the RHS in terms of momentum, Newton sez:

-kx=(dp/dt)

Now quantize:

-kx=(d/dt)(-i*hbar)(d/dx)

-(i*hbar)(d/dt)(d/dx)+kx=0

Now we have an operator. Let it act on a wavefunction Φ(x,t)

-(i*hbar)(d/dt)(dΦ(x,t)/dx)+kxΦ(x,t)=0

I just thought of this like 5 minutes ago, so I haven't worked it out yet. But what I want to explore is:

1. Is this mathematically valid?

2. If so, what is the relationship between this wavefunction and the Schrodinger wavefunction?

3. Has this been worked out before?

4. Might there be some advantage to this approach?

edit: fixed an omission, and a typo

Σ

**F**=m(d^{2}**x**/dt^{2})presupposed that we could know the trajectory

**x**(t) of a particle, which violates the uncertainty principle. However, lately I am thinking that that is not the reason at all. The Hamiltonian is expressed in terms of the coordinates and momenta, which we are also not allowed to know simultaneously.So why not try to quantize Newton?

I want to look at an example first, and then go into generalities later. Let's look at the simple harmonic oscillator in 1D along the x axis.

Writing the RHS in terms of momentum, Newton sez:

-kx=(dp/dt)

Now quantize:

-kx=(d/dt)(-i*hbar)(d/dx)

-(i*hbar)(d/dt)(d/dx)+kx=0

Now we have an operator. Let it act on a wavefunction Φ(x,t)

-(i*hbar)(d/dt)(dΦ(x,t)/dx)+kxΦ(x,t)=0

I just thought of this like 5 minutes ago, so I haven't worked it out yet. But what I want to explore is:

1. Is this mathematically valid?

2. If so, what is the relationship between this wavefunction and the Schrodinger wavefunction?

3. Has this been worked out before?

4. Might there be some advantage to this approach?

edit: fixed an omission, and a typo

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