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The simplest example of a quantum system with a position degree of freedom is a free particle in a single spatial dimension. A free particle is one which is not subject to external influences, so that its Hamiltonian consists only of its kinetic energy:

H=12mP2=−ℏ22md2dx2.
{\displaystyle H={\frac {1}{2m}}P^{2}=-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}.}

The general solution of the Schrödinger equation is given by

ψ(x,t)=12π∫−∞∞ψ^(k,0)ei(kx−ℏk22mt)dk,
{\displaystyle \psi (x,t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }{\hat {\psi }}(k,0)e^{i(kx-{\frac {\hbar k^{2}}{2m}}t)}\mathrm {d} k,}

which is a superposition of all possible plane waves ei(kx−ℏk22mt)
{\displaystyle e^{i(kx-{\frac {\hbar k^{2}}{2m}}t)}}
, which are eigenstates of the momentum operator with momentum p=ℏk
{\displaystyle p=\hbar k}
. The coefficients of the superposition are ψ^(k,0)
{\displaystyle {\hat {\psi }}(k,0)}
, which is the Fourier transform of the initial quantum state ψ(x,0)
{\displaystyle \psi (x,0)}
.
 
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Swede said:
The simplest example of a quantum system with a position degree of freedom is a free particle in a single spatial dimension. A free particle is one which is not subject to external influences, so that its Hamiltonian consists only of its kinetic energy:

H=12mP2=−ℏ22md2dx2.
{\displaystyle H={\frac {1}{2m}}P^{2}=-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}.}

The general solution of the Schrödinger equation is given by

ψ(x,t)=12π∫−∞∞ψ^(k,0)ei(kx−ℏk22mt)dk,
{\displaystyle \psi (x,t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }{\hat {\psi }}(k,0)e^{i(kx-{\frac {\hbar k^{2}}{2m}}t)}\mathrm {d} k,}

which is a superposition of all possible plane waves ei(kx−ℏk22mt)
{\displaystyle e^{i(kx-{\frac {\hbar k^{2}}{2m}}t)}}
, which are eigenstates of the momentum operator with momentum p=ℏk
{\displaystyle p=\hbar k}
. The coefficients of the superposition are ψ^(k,0)
{\displaystyle {\hat {\psi }}(k,0)}
, which is the Fourier transform of the initial quantum state ψ(x,0)
{\displaystyle \psi (x,0)}
.
Click to expand...
7 IQ
 
ILoveVesna said:
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Click to expand...
5 IQ
 
skibid ohio nigger! xX_skibidiRapist_Xx jarty;edo skibidi ohio yap dollar flicker munt aztec goonbumping while doing the polish wave

"Where's the punchline?"

THOSE WHO KNOW:
 

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Soyteen said:
skibid ohio nigger! xX_skibidiRapist_Xx jarty;edo skibidi ohio yap dollar flicker munt aztec goonbumping while doing the polish wave

"Where's the punchline?"

THOSE WHO KNOW:
Click to expand...
1 IQ
 

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