By A. B.; Krainov, V.; Leggett, Anthony J.(translator) Migdal

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**Additional info for Approximation Methods in Quantum Mechanics **

**Example text**

23. Putting it all together, we have shown that, if the matrix R is to preserve the normalization of probabilities, it must have the property that R† R = 1. 24) Matrices with this property are called unitary matrices. We have arrived at an important general fact: Any physically possible linear optical element in a two-beam interferometer is represented by a 2 × 2 unitary matrix. 14 Here is what we have proved: If R is to preserve the normalization of probabilities for any input v, then it must be unitary.

In general, then, the atom will be in a state |ψ = α0 |E0 + α1 |E1 . 48) The amplitudes αk = Ek |ψ . If the atom is in the state |ψ , then a measurement of its energy E will yield E0 with probability |α0 |2 and E1 with probability |α1 |2 . 25 Show that any state of the form eiφ |Ek is a state with deﬁnite energy Ek . The early quantum physicists thought that an atom must always be “in” one or another of its energy levels, and even today physicists, chemists, and others will often speak and write in this way.

However, they realized that the trajectory of an atom in a non-uniform magnetic ﬁeld depended on this energy. Consider an external magnetic z ﬁeld Bz that is increasing in the positive z-direction, so that dB dz > 0. The energy of a particle with a given μz will depend on the z-coordinate of its position: • If μz > 0, then E decreases as z increases: • If μz < 0, then E increases as z increases. This contributes to the effective potential energy for the atom, and so in either case there is a net force on it.

### Approximation Methods in Quantum Mechanics by A. B.; Krainov, V.; Leggett, Anthony J.(translator) Migdal

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