Raising & Lowering; Creating & Annihilating

Transcription

1 Raising & Lowering; Creating & Annihilating Frank Riou The purpose of this tutorial is to illustrate uses of the creation (raising) an annihilation (lowering) operators in the complementary coorinate an matri representations. These operators have routine utility in quantum mechanics in general, an are especially useful in the areas of quantum optics an quantum information. The harmonic oscillator eigenstates are regularly use to represent (in a ruimentary way) the vibrational states of iatomic molecules an also (more rigorously) the quantize states of the electromagnetic fiel. The creation operator as a quantum of energy to the molecule or the electromagnetic fiel an the annihilation operator oes the opposite. The harmonic oscillator eigenfunctions in coorinate space are given below, where v is the quantum number an can have the values,,,... ( v ) v v Her( v ep First we emonstrate that the harmonic oscillator eigenfunctions are normalize. ( ) ( ) ( ) Net we emonstrate that they are orthogonal: ( ) ( ) ( ) ( ) ( ) ( ) The harmonic oscillator eigenfunctions form an orthonormal basis set. They are isplaye below. ( ( ( The raising or creation operator in the coorinate representation in reuce units is the position operator minus i times the coorinate space momentum operator:

2 Operating on the v = eigenfunction yiels the v = eigenfunction: ( ( The lowering or annihilation operator in the coorinate representation in reuce units is the position operator plus i times the coorinate space momentum operator: Operating on the v = eigenfunction yiels the v = eigenfunction: ( ( 5 5 The energy operator in coorinate space an the energy epectation value for the v = state are given below. E v = v + / in atomic units. H = ( ) ( ) ( ).5 In the matri formulation of quantum mechanics the harmonic oscillator eigenfunctions are vectors. The matri representations for the v = state an the eigenstates use above are given below. They are actually infinite vectors which for practical purposes are truncate at orer 5. v v v v

3 They form an orthonormal basis set: v T v v T v v T v v T v v T v v T v In this contet the creation an annihilation operators are 55 matrices. Create 4 4 ˆ an nn The annihilation operator on the v = state:.44 The annihilation operator on the v = state: ˆ a n n n The creation operator on the v = state: Create.7 ˆ ˆ aan nn The number operator on the v = state: Create

4 Or o it this way: v T Create v v T Create v v T Create v v T Create v ˆ ˆ aa n n n The energy operator operating on the v = an 5 states: Create.5 Create 4.5 ˆ! n n a n Create Creating the v = eigenstate from the vacuum: Create v This operation is illustrate graphically in the coorinate representation as follows:

5 Construct the matri forms of the position an momentum operators using the annihilation an creation operators. See E. E. Anerson, Moern Physics an Quantum Mechanics, page. Position Create Momentum i Create ( ) Calculate the position an momentum epectation values for several states: v T Positionv v T Momentumv v T Positionv v T Momentumv Calculate the position momentum uncertainty prouct (p) for several states: v T Position v v T Positionv v T Momentum v v T Momentumv.5 v T Position v v T Positionv v T Momentum v v T Momentumv.5 Calculate the energy epectation value for the following superposition state. = v v v T Create 7 P E P E P E = 7 5 7

6 Below it is emonstrate that there are two equivalent forms of the harmonic oscillator energy operator in the matri formulation of quantum mechanics. ˆ ˆ aa n n n ˆˆ aa n n n Create.5 Create.5 Or, o it this way: Create Create.5 Or this way: Momentum Position.5 Demonstrate that the position an momentum operators onʹt commute using the matri form of the operators. i Momentum Position Position Momentum 4 This calculation yiels the ientity matri as epecte, ecept for the value of the last iagonal element. The latter is a mathematical artifact of using truncate matrices for operators which are infinite.