ℏ ≈ 0 and the spread of QM wave functionIs there a direct mathematical method to show that if a quantum wave funtion is initially sharply localized, then it will stay sharply localized if ℏ ≈ 0? In that case the Ehrenfest theorem implies the transition from quantum mechanics to classical mechanics.Of course, we are dealing with the propagation of a wave function, but let's not mess with path integrals. Thus, the structure of the general solution of Schrödinger equation should imply the result - if possible.

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ℏ  ≈  0 and the spread of QM wave functionIs there a direct mathematical method to show that if a quantum wave funtion is initially sharply localized, then it will stay sharply localized if   ℏ  ≈  0? In that case the Ehrenfest theorem implies the transition from quantum mechanics to classical mechanics.Of course, we are dealing with the propagation of a wave function, but let&#039;s not mess with path integrals. Thus, the structure of the general solution of Schrödinger equation should imply the result - if possible.

Mekjulleymg · Accepted Answer

The propagator for a one-dimensional free particle is, for example,  K  (      x    ′    −  x  ,      t    ′    −  t  )  =            m              2        π                  i                ℏ        (                  t          ′                −        t        )              exp  ⁡      (    −                  m        (                  x          ′                −        x                  )          2                            2        π                  i                ℏ        (                  t          ′                −        t        )              )    .Meaning that   ψ  (      x    ′    ,      t    ′    )  =  ∫  K  (      x    ′    −  x  ,      t    ′    −  t  )    ψ  (  x  ,  t  )        d    x. Here,   m is the particle mass.In plain(er) English, time evolution of a free particle comes down to a convolution with a Gaussian kernel with width proportional to   ℏ. Mathematically, you can take   ℏ  →  0, the Gaussian function becomes a   δ function and the wave function does not spread any more. Physically that means that the spread is negligible if   ℏ is very small compared to             m              L        2              T  , where   L and   T are the physically relevant length and time scales, respectively.If you want to take the classical limit properly, you need decoherence, as discussed in other answers.

ttyme411gl · Accepted Answer

The problem with trying to understand the spread of the wavefunction in the classical limit by taking   ℏ  ≈  0 or taking the limit   ℏ  →  0 is that in reality   ℏ  ≠  0. Taking the real non-zero value of   ℏthere are cases of perfectly ordinary objects for which Ehrenfest&#039;s theorem doesn&#039;t imply anything like classical behaviour. Rather the wavefunction of that object spreads out a lot over time. The classical limit is actually a result of decoherence and information being copied out of &#039;classical&#039; objects into the environment.

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