慕容青草

今天有典型的中式网络思维者给我短信号称他知道如何推导薛定谔方程。原本对这种中式网络专业人士的标致性的轻佻思维应该置之一笑而已，但正好在网上碰上一则讲解为什么我们无法推导薛定谔方程的帖子，就将之转贴于此 （https://www.quora.com/Why-we-say-that-we-cant-derive-Schrodinger-equation）：

Why we say that we can't "derive" Schrodinger equation?

Schrodinger's equation cannot be derived. It was thought up using logical arguments and so far it has seemed to work experimentally.

The equations is essentially a re-write up for energy conservation:

E=T+V

Where T

is the Kinetic Energy and V

is the potential. However, to be more explicit we must work with operators (if you are unsure what operators are I suggest you look them up; this will give you a better understanding of what's going on).

The KE for a particle is given by the KE Operator:

T^=−ℏ22m∂2∂x2

This comes from the momentum operator of the particle/wave p^=−ih∂/∂x

. You use this in the analogous classical mechanics equation for KE to obtain T^

(Try doing this as an exercise).

So now we are left with just putting it all together. The first equation turns into:

−ℏ22m∂2Ψ∂x2+V(x)Ψ=EΨ

And we define the Hamiltonian H^ as:

H^=−ℏ22m∂2∂x2+V(x)

Thus:

H^Ψ=EΨ

注：其实我们之所以不能推导薛定谔方程的原因比此贴所解释更深刻。我在给该网络思维人士的回帖中对其背后的哲学原因进行了解释。这里就不赘述。上面这个短贴已经把基本的要点做了一些解释。