Note 2

05-01

II. Quantum Mechanics 2

An interesting and pedagogical model to study is harmonic oscillators, which, could be considered as the origin of (what I have learned and understood) quantum field theory, where every position on 4-d spacetime is given by several fields which each could be expanded in a form of harmonic oscillator.

Classically, harmonic oscillator is interesting for its simple analytical solution and constant period. Its energy is then a function of amplitude. In the case of (bosonic) harmonic oscillators, its a little bit different for the energy interpretation is of the number of particles (what is called second quantization approach). Semi-classically, one would also pursue the origin of wave function in a harmonic potential, which is pretty much a like classical oscillators. But within this kind of consideration, the basic and framework would be tricky and vague.

The start point is the Hamiltonian that $H=frac { 1 }{ 2 } m omega^{ 2 } x^{ 2 } + frac { 1 }{ 2m } p^{ 2 }$ . Noticing the commutator between the two operators $[x, p] = i hbar o i$ , one would prefer to express the Hamiltonian into a non-negative form, where the operator $a = sqrt {frac{ m omega }{ 2 }} left ( x + frac{ ip } { m omega } ight )$ and its hermitian conjugate $a^{ dagger }$ is introduced naturally. Therefore, the Hamiltonian is $H = omega left ( frac{ 1 }{ 2 } + a^{dagger} a ight )$ . The non-negativity is proved directly due to $langle psi | a^{dagger} a | psi angle = ||a | psi angle ||^2$ for $langle psi | a^{ dagger } = ( a | psi angle ) ^{ dagger }$ . The vacuum is simple to define where $a | ext{vacuum} angle = 0$ , where its coordinate wave function $psi_{ ext{vac} } ( x ) = langle x | ext{vac} angle$ . By taking $langle {f x} | a | ext{vac} angle = left langle {f x} middle | sqrt {frac{ m omega }{ 2 }} left ( x + frac{ ip } { m omega } ight ) middle | ext{vac} ight angle = 0$ . Here, the notation is a little bit noisy, for $f x$ is a real parameter, and $x$ is for operator, i.e., $hat{ x }$ . By substituting $p o -i partial_{ x }$ , which is derived whenever the commutator is introduced (see Sakurai as I introduced in previous note), one gets $left ( {f x} + frac{ 1 }{ m omega } frac{ d }{ d{f x} } ight ) langle {f x} | ext{vac} angle = 0$ which eventually gives $langle {f x} | ext{vac} angle = left ( frac{ 1 }{ pi^{ 1 / 4 } ( m omega )^{ 1 / 4 } } ight ) exp left ( - frac{ 1 }{ 2 } m omega {f x}^{2} ight )$ . This form has a good plot profile, the Gaussian shape. The physical interpretation is its ground state is mainly localized in the center of potential.

This is a brief introduction to the harmonic oscillator. The following example is what Im going to talk, as an example of quantum mechanics that I have introduced so far: a linear shift term (constant force field) in harmonic oscillator. $H = frac { 1 }{ 2 } m omega^{ 2 } x^{ 2 } + frac { 1 }{ 2m } p^{ 2 } + mgx$ . Classically we know it is nothing but a shift of origin point, and quantum mechanically we can do it by changing the coordinate ${f x} o {f x} + frac{ g }{ omega^{ 2 } }$ , but it would be of interest to treat it under the basis of origin solution of $H = frac { 1 }{ 2 } m omega^{ 2 } x^{ 2 } + frac { 1 }{ 2m } p^{ 2 }$ . So the following would be a start point: find the evolution of state whose initial value is vacuum of origin Hamiltonian while at $t = 0$ the Hamiltonian suddenly contains a shift term.

The solution strongly correspondes to the classical solution, which gives what usually called physics picture clear in quantum mechanics. I will talk into the details in next note. It will be released soon, maybe 2 to 4 days.

A special technique to solve this problem is considering a special operator $exp left ( sqrt{ frac{ mg^{ 2 } }{ 2 omega^{ 3 } } } ( a^{ dagger } - a ) ight )$ . Without this, it can also be solved with some tedious calculation.