### 3.3 The Op­er­a­tors of Quan­tum Me­chan­ics

The nu­mer­i­cal quan­ti­ties that the old New­ton­ian physics uses, (po­si­tion, mo­men­tum, en­ergy, ...), are just shad­ows of what re­ally de­scribes na­ture: op­er­a­tors. The op­er­a­tors de­scribed in this sec­tion are the key to quan­tum me­chan­ics.

As the first ex­am­ple, while a math­e­mat­i­cally pre­cise value of the po­si­tion of a par­ti­cle never ex­ists, in­stead there is an -​po­si­tion op­er­a­tor . It turns the wave func­tion into :

 (3.3)

The op­er­a­tors and are de­fined sim­i­larly as .

In­stead of a lin­ear mo­men­tum , there is an -​mo­men­tum op­er­a­tor

 (3.4)

that turns into its -​de­riv­a­tive:
 (3.5)

The con­stant is called “Planck's con­stant.” (Or rather, it is Planck's orig­i­nal con­stant di­vided by .) If it would have been zero, all these trou­bles with quan­tum me­chan­ics would not oc­cur. The blobs would be­come points. Un­for­tu­nately, is very small, but nonzero. It is about 10 kg m/s.

The fac­tor in makes it a Her­mit­ian op­er­a­tor (a proof of that is in de­riva­tion {D.9}). All op­er­a­tors re­flect­ing macro­scopic phys­i­cal quan­ti­ties are Her­mit­ian.

The op­er­a­tors and are de­fined sim­i­larly as :

 (3.6)

The ki­netic en­ergy op­er­a­tor is:

 (3.7)

Its shadow is the New­ton­ian no­tion that the ki­netic en­ergy equals:

This is an ex­am­ple of the “New­ton­ian anal­ogy”: the re­la­tion­ships be­tween the dif­fer­ent op­er­a­tors in quan­tum me­chan­ics are in gen­eral the same as those be­tween the cor­re­spond­ing nu­mer­i­cal val­ues in New­ton­ian physics. But since the mo­men­tum op­er­a­tors are gra­di­ents, the ac­tual ki­netic en­ergy op­er­a­tor is, from the mo­men­tum op­er­a­tors above:
 (3.8)

Math­e­mati­cians call the set of sec­ond or­der de­riv­a­tive op­er­a­tors in the ki­netic en­ergy op­er­a­tor the Lapla­cian, and in­di­cate it by :

 (3.9)

In those terms, the ki­netic en­ergy op­er­a­tor can be writ­ten more con­cisely as:
 (3.10)

Fol­low­ing the New­ton­ian anal­ogy once more, the to­tal en­ergy op­er­a­tor, in­di­cated by , is the the sum of the ki­netic en­ergy op­er­a­tor above and the po­ten­tial en­ergy op­er­a­tor :

 (3.11)

This to­tal en­ergy op­er­a­tor is called the Hamil­ton­ian and it is very im­por­tant. Its eigen­val­ues are in­di­cated by (for en­ergy), for ex­am­ple , , , ...with:

 (3.12)

where is eigen­func­tion num­ber of the Hamil­ton­ian.

It is seen later that in many cases a more elab­o­rate num­ber­ing of the eigen­val­ues and eigen­vec­tors of the Hamil­ton­ian is de­sir­able in­stead of us­ing a sin­gle counter . For ex­am­ple, for the elec­tron of the hy­dro­gen atom, there is more than one eigen­func­tion for each dif­fer­ent eigen­value , and ad­di­tional coun­ters and are used to dis­tin­guish them. It is usu­ally best to solve the eigen­value prob­lem first and de­cide on how to num­ber the so­lu­tions af­ter­wards.

(It is also im­por­tant to re­mem­ber that in the lit­er­a­ture, the Hamil­ton­ian eigen­value prob­lem is com­monly re­ferred to as the time-in­de­pen­dent Schrö­din­ger equa­tion. How­ever, this book prefers to re­serve the term Schrö­din­ger equa­tion for the un­steady evo­lu­tion of the wave func­tion.)

Key Points
Phys­i­cal quan­ti­ties cor­re­spond to op­er­a­tors in quan­tum me­chan­ics.

Ex­pres­sions for var­i­ous im­por­tant op­er­a­tors were given.

Ki­netic en­ergy is in terms of the so-called Lapla­cian op­er­a­tor.

The im­por­tant to­tal en­ergy op­er­a­tor, (ki­netic plus po­ten­tial en­ergy,) is called the Hamil­ton­ian.