# 态叠加原理本文重定向自 态叠加原理

## 理论

### 电子自旋范例

${\displaystyle |\psi \rangle =c_{\uparrow }|\uparrow \rangle +c_{\downarrow }|\downarrow \rangle }$

${\displaystyle p_{\uparrow }=|c_{\uparrow }|^{2}}$
${\displaystyle p_{\downarrow }=|c_{\downarrow }|^{2}}$

${\displaystyle |\psi \rangle ={3i \over 5}|\uparrow \rangle +{4 \over 5}|\downarrow \rangle }$

${\displaystyle p_{\uparrow }=\left|\;{\frac {3i}{5}}\;\right|^{2}={\frac {9}{25}}}$
${\displaystyle p_{\downarrow }=\left|\;{\frac {4}{5}}\;\right|^{2}={\frac {16}{25}}}$

${\displaystyle p={\frac {9}{25}}+{\frac {16}{25}}=1}$

### 非相对论性自由粒子案例

${\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\ \Psi (\mathbf {r} ,t)=i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,t)}$

${\displaystyle \Psi (\mathbf {r} ,t)=e^{i(\mathbf {k} \cdot \mathbf {r} -\omega t)}}$

${\displaystyle {\frac {\hbar ^{2}k^{2}}{2m}}=\hbar \omega }$

${\displaystyle \Psi (\mathbf {r} ,t)={\frac {1}{(2\pi )^{3/2}}}\int _{\mathbb {K} }A(\mathbf {k} )e^{i(\mathbf {k} \cdot \mathbf {r} -\omega t)}\mathrm {d} \mathbf {k} }$

${\displaystyle \Psi (x,t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }A(k)~e^{i(kx-\omega (k)t)}\ \mathrm {d} k}$

${\displaystyle A(k)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\,\infty }\Psi (x,0)~e^{-ikx}\,\mathrm {d} x}$

## 参考文献

1. French, Anthony, An Introduction to Quantum Physics, W. W. Norton, Inc., 1978, ISBN 0-393-09106-0 请检查|isbn=值 (帮助)
• Bohr, N. (1927/1928). The quantum postulate and the recent development of atomic theory, Nature Supplement 14 April 1928, 121: 580–590页面存档备份，存于互联网档案馆）.
• Cohen-Tannoudji, C., Diu, B., Laloë, F. (1973/1977). Quantum Mechanics, translated from the French by S. R. Hemley, N. Ostrowsky, D. Ostrowsky, second edition, volume 1, Wiley, New York, ISBN 0471164321.
• Dirac, P. A. M. (1930/1958). The Principles of Quantum Mechanics, 4th edition, Oxford University Press.
• Einstein, A. (1949). Remarks concerning the essays brought together in this co-operative volume, translated from the original German by the editor, pp. 665–688 in Schilpp, P. A. editor (1949), Albert Einstein: Philosopher-Scientist, volume II, Open Court, La Salle IL.
• Feynman, R. P., Leighton, R.B., Sands, M. (1965). The Feynman Lectures on Physics, volume 3, Addison-Wesley, Reading, MA.
• Merzbacher, E. (1961/1970). Quantum Mechanics, second edition, Wiley, New York.
• Messiah, A. (1961). Quantum Mechanics, volume 1, translated by G.M. Temmer from the French Mécanique Quantique, North-Holland, Amsterdam.
• Wheeler, J. A.; Zurek, W.H. Quantum Theory and Measurement. Princeton NJ: Princeton University Press. 1983. 已忽略文本“John Archibald Wheeler” (帮助)