There is some confusion regarding the relationship between the raising and lowering ladder operators and the creation and annihilation operators commonly used in quantum field theory. The creation operator ai† increments the number of particles in state i, while the corresponding annihilation operator ai decrements the number of particles in state i. This clearly satisfies the requirements of the above definition of a ladder operator: the incrementing or decrementing of the eigenvalue of another operator (in this case the particle number operator).
Confusion arises because the term ladder operator is typically used to describe an operator that acts to increment or decrement a quantum number describing the state of a system. To change the state of a particle with the creation/annihilation operators of QFT requires the use of both annihilation and creation operators. An annihilation operator is used to remove a particle from the initial state and a creation operator is used to add a particle to the final state.
The term "ladder operator" is also sometimes used in mathematics, in the context of the theory of Lie algebras and in particular the affine Lie algebras, to describe the su(2) subalgebras, from which the root system and the highest weight modules can be constructed by means of the ladder operators.[1] In particular, the highest weight is annihilated by the raising operators; the rest of the positive root space is obtained by repeatedly applying the lowering operators (one set of ladder operators per subalgebra).
for some scalar c. If is an eigenstate of N with eigenvalue equation,
then the operator X acts on in such a way as to shift the eigenvalue by c:
In other words, if is an eigenstate of N with eigenvalue n then is an eigenstate of N with eigenvalue n + c or it is zero. The operator X is a raising operator for N if c is real and positive, and a lowering operator for N if c is real and negative.
A particular application of the ladder operator concept is found in the quantum mechanical treatment of angular momentum. For a general angular momentum vector, J, with components, Jx, Jy and Jz one defines the two ladder operators, J+ and J–,[2]
where εijk is the Levi-Civita symbol and each of i, j and k can take any of the values x, y and z.
From this, the commutation relations among the ladder operators and Jz are obtained,
(Technically, this is the Lie algebra of ).
The properties of the ladder operators can be determined by observing how they modify the action of the Jz operator on a given state,
Compare this result with
Thus one concludes that is some scalar multiplied by ,
This illustrates the defining feature of ladder operators in quantum mechanics: the incrementing (or decrementing) of a quantum number, thus mapping one quantum state onto another. This is the reason that they are often known as raising and lowering operators.
To obtain the values of α and β first take the norm of each operator, recognizing that J+ and J− are a Hermitian conjugate pair (),
The product of the ladder operators can be expressed in terms of the commuting pair J2 and Jz,
Thus, one may express the values of |α|2 and |β|2 in terms of the eigenvalues of J2 and Jz,
The angular momentum algebra can often be simplified by recasting it in the spherical basis. Using the notation of spherical tensor operators, the "-1", "0" and "+1" components of J(1) ≡ J are given by,[5]
From these definitions, it can be shown that the above scalar product can be expanded as
The significance of this expansion is that it clearly indicates which states are coupled by this term in the Hamiltonian, that is those with quantum numbers differing by mi = ±1 and mj = ∓1 only.
Another application of the ladder operator concept is found in the quantum mechanical treatment of the harmonic oscillator. We can define the lowering and raising operators as
They provide a convenient means to extract energy eigenvalues without directly solving the system's differential equation.
There are two main approaches given in the literature using ladder operators, one using the Laplace–Runge–Lenz vector, another using factorization of the Hamiltonian.
Laplace–Runge–Lenz vector
Another application of the ladder operator concept is found in the quantum mechanical treatment of the electronic energy of hydrogen-like atoms and ions. The Laplace–Runge–Lenz vector commutes with the Hamiltonian for an inverse square spherically symmetric potential and can be used to determine ladder operators for this potential.[6][7] We can define the lowering and raising operators (based on the classical Laplace–Runge–Lenz vector)
where is the angular momentum, is the linear momentum, is the reduced mass of the system, is the electronic charge, and is the atomic number of the nucleus. Analogous to the angular momentum ladder operators, one has and .
The commutators needed to proceed are:
and
. Therefore,
and
so
where the "?" indicates a nascent quantum number which emerges from the discussion.
Given the Pauli[8][9] equations Pauli Equation IV:
and Pauli Equation III:
and starting with the equation
and expanding, one obtains (assuming is the maximum value of the angular momentum quantum number consonant with all other conditions),
implying that , where is the traditional quantum number.
Factorization of the Hamiltonian
The Hamiltonian for a hydrogen-like potential can be written in spherical coordinates as
where and is the radial momentum
which is real and self-conjugate.
Suppose is an eigenvector of the Hamiltonian where is the angular momentum and represents the energy, so and we may label the Hamiltonian as
The factorization method was developed by Infeld and Hull[10] for differential equations. Newmarch and Golding[11] applied it to spherically symmetric potentials using operator notation.
Suppose we can find a factorization of the Hamiltonian by operators as
(1)
and
for scalars and . The vector may be evaluated in two different ways as
which can be re-arranged as
showing that is an eigenstate of with eigenvalue
If then and the states and have the same energy.
For the hydrogenic atom, setting
with
a suitable equation for is
with
There is an upper bound to the ladder operator if the energy is negative, (so for some ) then from Equation (1)
and can be identified with
Relation to group theory
Whenever there is degeneracy in a system, there is usually a related symmetry property and group. The degeneracy of the energy levels for the same value of but different angular momenta has been identified as the SO(4) symmetry of the spherically symmetric Coulomb potential.[12][13]
Now the Hamiltonian only has positive energy levels as can be seen from
This means that for some value of the series must terminate with and then
This is decreasing in energy by unless for some value of , . Identifying this value as gives
It then follows the so that
giving a recursion relation on with solution
There is degeneracy caused from angular momentum; there is additional degeneracy caused by the oscillator potential. Consider the states and apply the lowering operators : giving the sequence with the same energy but with decreasing by 2. In addition to the angular momentum degeneracy, this gives a total degeneracy of [14]
Relation to group theory
The degeneracies of the 3D isotropic harmonic oscillator are related to the special unitary group SU(3)[14][15]
History
Many sources credit Dirac with the invention of ladder operators.[16] Dirac's use of the ladder operators shows that the total angular momentum quantum number needs to be a non-negative half integer multiple of ħ.
^Fuchs, Jurgen (1992), Affine Lie Algebras and Quantum Groups, Cambridge University Press, ISBN0-521-48412-X
^de Lange, O. L.; R. E. Raab (1986). "Ladder operators for orbital angular momentum". American Journal of Physics. 54 (4): 372–375. Bibcode:1986AmJPh..54..372D. doi:10.1119/1.14625.
^Woodgate, Gordon K. (1983-10-06). Elementary Atomic Structure. ISBN978-0-19-851156-4. Retrieved 2009-03-03.
^"Angular Momentum Operators". Graduate Quantum Mechanics Notes. University of Virginia. Retrieved 2009-04-06.
^David, C. W. (1966). "Ladder Operator Solution for the Hydrogen Atom Electronic Energy Levels". American Journal of Physics. 34 (10): 984–985. Bibcode:1966AmJPh..34..984D. doi:10.1119/1.1972354.
^Burkhardt, C. E.; Levanthal, J. (2004). "Lenz vector operations on spherical hydrogen atom eigenfunctions". American Journal of Physics. 72 (8): 1013–1016. Bibcode:2004AmJPh..72.1013B. doi:10.1119/1.1758225.
^Pauli, Wolfgang (1926). "Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik". Z. Phys. 36 (5): 336–363. Bibcode:1926ZPhy...36..336P. doi:10.1007/BF01450175. S2CID 128132824.
^ B. L. Van der Waerden, Sources of Quantum Mechanics, Dover, New York, 1968
^ ab Newmarch, J. D.; Golding, R. M. (1978). "Ladder operators for some spherically symmetric potentials in quantum". Am. J. Phys. 46: 658–660. doi:10.1119/1.11225.
^Weinberg, S. J. (2011). "The SO(4) Symmetry of the Hydrogen Atom" (PDF). {{cite journal}}: Cite journal requires |journal= (help)
^ Lahiri, A.; Roy, P. K.; Bagchi, B. (1989). "Supersymmetry and the Ladder Operator Technique in Quantum Mechanics: The Radial Schrödinger Equation". Int. J. Theor. Phys. 28 (2): 183–189. Bibcode:1989IJTP...28..183L. doi:10.1007/BF00669809. S2CID 123255435.
^ abKirson, M. W. (2013). "Introductory Algebra for Physicists: Isotropic harmonic oscillator" (PDF). Weizmann Institute of Science. Retrieved 28 July 2021.
^ , D. M. "." 33 (3) (1965) 207–211. Fradkin, D. M. (1965). "Three-dimensional isotropic harmonic oscillator and SU3". Am. J. Phys. 33 (3): 207–211. Bibcode:1965AmJPh..33..207F. doi:10.1119/1.1971373.
ladder, operator, linear, algebra, application, quantum, mechanics, raising, lowering, operator, collectively, known, ladder, operators, operator, that, increases, decreases, eigenvalue, another, operator, quantum, mechanics, raising, operator, sometimes, call. In linear algebra and its application to quantum mechanics a raising or lowering operator collectively known as ladder operators is an operator that increases or decreases the eigenvalue of another operator In quantum mechanics the raising operator is sometimes called the creation operator and the lowering operator the annihilation operator Well known applications of ladder operators in quantum mechanics are in the formalisms of the quantum harmonic oscillator and angular momentum Contents 1 Terminology 2 General formulation 3 Angular momentum 3 1 Applications in atomic and molecular physics 4 Harmonic oscillator 5 Hydrogen like atom 5 1 Laplace Runge Lenz vector 5 2 Factorization of the Hamiltonian 5 3 Relation to group theory 6 3D isotropic harmonic oscillator 6 1 Factorization method 6 2 Relation to group theory 7 History 8 See also 9 ReferencesTerminology EditMain article Creation and annihilation operators There is some confusion regarding the relationship between the raising and lowering ladder operators and the creation and annihilation operators commonly used in quantum field theory The creation operator ai increments the number of particles in state i while the corresponding annihilation operator ai decrements the number of particles in state i This clearly satisfies the requirements of the above definition of a ladder operator the incrementing or decrementing of the eigenvalue of another operator in this case the particle number operator Confusion arises because the term ladder operator is typically used to describe an operator that acts to increment or decrement a quantum number describing the state of a system To change the state of a particle with the creation annihilation operators of QFT requires the use of both annihilation and creation operators An annihilation operator is used to remove a particle from the initial state and a creation operator is used to add a particle to the final state The term ladder operator is also sometimes used in mathematics in the context of the theory of Lie algebras and in particular the affine Lie algebras to describe the su 2 subalgebras from which the root system and the highest weight modules can be constructed by means of the ladder operators 1 In particular the highest weight is annihilated by the raising operators the rest of the positive root space is obtained by repeatedly applying the lowering operators one set of ladder operators per subalgebra General formulation EditSuppose that two operators X and N have the commutation relation N X c X displaystyle N X cX for some scalar c If n displaystyle n rangle is an eigenstate of N with eigenvalue equation N n n n displaystyle N n rangle n n rangle then the operator X acts on n displaystyle n rangle in such a way as to shift the eigenvalue by c N X n X N N X n X N n N X n X n n c X n n c X n displaystyle begin aligned NX n rangle amp XN N X n rangle amp XN n rangle N X n rangle amp Xn n rangle cX n rangle amp n c X n rangle end aligned In other words if n displaystyle n rangle is an eigenstate of N with eigenvalue n then X n displaystyle X n rangle is an eigenstate of N with eigenvalue n c or it is zero The operator X is a raising operator for N if c is real and positive and a lowering operator for N if c is real and negative If N is a Hermitian operator then c must be real and the Hermitian adjoint of X obeys the commutation relation N X c X displaystyle N X dagger cX dagger In particular if X is a lowering operator for N then X is a raising operator for N and vice versa Angular momentum EditMain article Angular momentum operator A particular application of the ladder operator concept is found in the quantum mechanical treatment of angular momentum For a general angular momentum vector J with components Jx Jy and Jz one defines the two ladder operators J and J 2 J J x i J y displaystyle J J x iJ y J J x i J y displaystyle J J x iJ y where i is the imaginary unit The commutation relation between the cartesian components of any angular momentum operator is given by J i J j i ℏ ϵ i j k J k displaystyle J i J j i hbar epsilon ijk J k where eijk is the Levi Civita symbol and each of i j and k can take any of the values x y and z From this the commutation relations among the ladder operators and Jz are obtained J z J ℏ J displaystyle left J z J pm right pm hbar J pm J J 2 ℏ J z displaystyle left J J right 2 hbar J z Technically this is the Lie algebra of s l 2 R displaystyle mathfrak s l 2 mathbb R The properties of the ladder operators can be determined by observing how they modify the action of the Jz operator on a given state J z J j m J J z J z J j m J J z ℏ J j m ℏ m 1 J j m displaystyle begin aligned J z J pm j m rangle amp left J pm J z left J z J pm right right j m rangle amp left J pm J z pm hbar J pm right j m rangle amp hbar left m pm 1 right J pm j m rangle end aligned Compare this result withJ z j m 1 ℏ m 1 j m 1 displaystyle J z j m pm 1 rangle hbar m pm 1 j m pm 1 rangle Thus one concludes that J j m displaystyle J pm j m rangle is some scalar multiplied by j m 1 displaystyle j m pm 1 rangle J j m a j m 1 displaystyle J j m rangle alpha j m 1 rangle J j m b j m 1 displaystyle J j m rangle beta j m 1 rangle This illustrates the defining feature of ladder operators in quantum mechanics the incrementing or decrementing of a quantum number thus mapping one quantum state onto another This is the reason that they are often known as raising and lowering operators To obtain the values of a and b first take the norm of each operator recognizing that J and J are a Hermitian conjugate pair J J displaystyle J pm J mp dagger j m J J j m j m J J j m j m 1 a a j m 1 a 2 displaystyle langle j m J dagger J j m rangle langle j m J J j m rangle langle j m 1 alpha alpha j m 1 rangle alpha 2 j m J J j m j m J J j m j m 1 b b j m 1 b 2 displaystyle langle j m J dagger J j m rangle langle j m J J j m rangle langle j m 1 beta beta j m 1 rangle beta 2 The product of the ladder operators can be expressed in terms of the commuting pair J2 and Jz J J J x i J y J x i J y J x 2 J y 2 i J x J y J 2 J z 2 ℏ J z displaystyle J J J x iJ y J x iJ y J x 2 J y 2 i J x J y J 2 J z 2 hbar J z J J J x i J y J x i J y J x 2 J y 2 i J x J y J 2 J z 2 ℏ J z displaystyle J J J x iJ y J x iJ y J x 2 J y 2 i J x J y J 2 J z 2 hbar J z Thus one may express the values of a 2 and b 2 in terms of the eigenvalues of J2 and Jz a 2 ℏ 2 j j 1 ℏ 2 m 2 ℏ 2 m ℏ 2 j m j m 1 displaystyle alpha 2 hbar 2 j j 1 hbar 2 m 2 hbar 2 m hbar 2 j m j m 1 b 2 ℏ 2 j j 1 ℏ 2 m 2 ℏ 2 m ℏ 2 j m j m 1 displaystyle beta 2 hbar 2 j j 1 hbar 2 m 2 hbar 2 m hbar 2 j m j m 1 The phases of a and b are not physically significant thus they can be chosen to be positive and real Condon Shortley phase convention We then have 3 J j m ℏ j m j m 1 j m 1 ℏ j j 1 m m 1 j m 1 displaystyle J j m rangle hbar sqrt j m j m 1 j m 1 rangle hbar sqrt j j 1 m m 1 j m 1 rangle J j m ℏ j m j m 1 j m 1 ℏ j j 1 m m 1 j m 1 displaystyle J j m rangle hbar sqrt j m j m 1 j m 1 rangle hbar sqrt j j 1 m m 1 j m 1 rangle Confirming that m is bounded by the value of j j m j displaystyle j leq m leq j one hasJ j j 0 displaystyle J j j rangle 0 J j j 0 displaystyle J j j rangle 0 The above demonstration is effectively the construction of the Clebsch Gordan coefficients Applications in atomic and molecular physics Edit Many terms in the Hamiltonians of atomic or molecular systems involve the scalar product of angular momentum operators An example is the magnetic dipole term in the hyperfine Hamiltonian 4 H D A I J displaystyle hat H text D hat A mathbf I cdot mathbf J where I is the nuclear spin The angular momentum algebra can often be simplified by recasting it in the spherical basis Using the notation of spherical tensor operators the 1 0 and 1 components of J 1 J are given by 5 J 1 1 1 2 J x i J y J 2 J 0 1 J z J 1 1 1 2 J x i J y J 2 displaystyle begin aligned J 1 1 amp dfrac 1 sqrt 2 J x iJ y dfrac J sqrt 2 J 0 1 amp J z J 1 1 amp frac 1 sqrt 2 J x iJ y frac J sqrt 2 end aligned From these definitions it can be shown that the above scalar product can be expanded asI 1 J 1 n 1 1 1 n I n 1 J n 1 I 0 1 J 0 1 I 1 1 J 1 1 I 1 1 J 1 1 displaystyle mathbf I 1 cdot mathbf J 1 sum n 1 1 1 n I n 1 J n 1 I 0 1 J 0 1 I 1 1 J 1 1 I 1 1 J 1 1 The significance of this expansion is that it clearly indicates which states are coupled by this term in the Hamiltonian that is those with quantum numbers differing by mi 1 and mj 1 only Harmonic oscillator EditMain article Quantum harmonic oscillator Another application of the ladder operator concept is found in the quantum mechanical treatment of the harmonic oscillator We can define the lowering and raising operators asa m w 2 ℏ x i m w p a m w 2 ℏ x i m w p displaystyle begin aligned hat a amp sqrt m omega over 2 hbar left hat x i over m omega hat p right hat a dagger amp sqrt m omega over 2 hbar left hat x i over m omega hat p right end aligned They provide a convenient means to extract energy eigenvalues without directly solving the system s differential equation Hydrogen like atom EditMain article Hydrogen like atom There are two main approaches given in the literature using ladder operators one using the Laplace Runge Lenz vector another using factorization of the Hamiltonian Laplace Runge Lenz vector Edit Another application of the ladder operator concept is found in the quantum mechanical treatment of the electronic energy of hydrogen like atoms and ions The Laplace Runge Lenz vector commutes with the Hamiltonian for an inverse square spherically symmetric potential and can be used to determine ladder operators for this potential 6 7 We can define the lowering and raising operators based on the classical Laplace Runge Lenz vector A 1 Z e 2 m L p i ℏ p r r displaystyle vec A left frac 1 Ze 2 mu right left vec L times vec p boldsymbol i hbar vec p right frac vec r r where L displaystyle vec L is the angular momentum p displaystyle vec p is the linear momentum m displaystyle mu is the reduced mass of the system e displaystyle e is the electronic charge and Z displaystyle Z is the atomic number of the nucleus Analogous to the angular momentum ladder operators one has A A x i A y displaystyle A A x iA y and A A x i A y displaystyle A A x iA y The commutators needed to proceed are A L z i ℏ A displaystyle A pm L z mp boldsymbol i hbar A mp and A L 2 2 ℏ 2 A 2 ℏ A L z 2 ℏ A z L displaystyle A pm L 2 mp 2 hbar 2 A pm 2 hbar A pm L z pm 2 hbar A z L pm Therefore A ℓ m ℓ ℓ m ℓ 1 displaystyle A ell m ell rangle rightarrow ell m ell 1 rangle and L 2 A ℓ ℓ ℏ 2 ℓ 1 ℓ 1 1 A ℓ ℓ displaystyle L 2 left A ell ell rangle right hbar 2 ell 1 ell 1 1 left A ell ell rangle right so A ℓ ℓ ℓ 1 ℓ 1 displaystyle A ell ell rangle rightarrow ell 1 ell 1 rangle where the indicates a nascent quantum number which emerges from the discussion Given the Pauli 8 9 equations Pauli Equation IV 1 A A 2 E m Z 2 e 4 L 2 ℏ 2 displaystyle 1 A cdot A left frac 2E mu Z 2 e 4 right L 2 hbar 2 and Pauli Equation III A A j 2 i ℏ E m Z 2 e 4 L j displaystyle left A times A right j left frac 2 boldsymbol i hbar E mu Z 2 e 4 right L j and starting with the equation A A ℓ ℓ 0 displaystyle A A ell ell rangle 0 and expanding one obtains assuming ℓ displaystyle ell is the maximum value of the angular momentum quantum number consonant with all other conditions 1 2 E m Z 2 e 4 L 2 ℏ 2 i 2 i ℏ E m Z 2 e 4 L z ℓ ℓ 0 displaystyle left 1 frac 2E mu Z 2 e 4 L 2 hbar 2 i frac 2i hbar E mu Z 2 e 4 L z right ell ell rangle 0 which leads to the Rydberg formula E n m Z 2 e 4 2 ℏ 2 ℓ 1 2 displaystyle E n frac mu Z 2 e 4 2 hbar 2 ell 1 2 implying that ℓ 1 n displaystyle ell 1 n where n displaystyle n is the traditional quantum number Factorization of the Hamiltonian Edit The Hamiltonian for a hydrogen like potential can be written in spherical coordinates asH 1 2 m p r 2 1 r 2 L 2 V r displaystyle H 1 2 mu p r 2 1 r 2 L 2 V r where V r Z e 2 r displaystyle V r Ze 2 r and p r displaystyle p r is the radial momentum p r x r p x y r p y z r p z displaystyle p r x r p x y r p y z r p z which is real and self conjugate Suppose n l displaystyle nl rangle is an eigenvector of the Hamiltonian where l displaystyle l rangle is the angular momentum and n displaystyle n represents the energy so L 2 n l l l 1 ℏ 2 n l displaystyle L 2 nl rangle l l 1 hbar 2 nl rangle and we may label the Hamiltonian as H l displaystyle H l H 1 2 m p r 2 1 r 2 l l 1 ℏ 2 V r displaystyle H 1 2 mu p r 2 1 r 2 l l 1 hbar 2 V r The factorization method was developed by Infeld and Hull 10 for differential equations Newmarch and Golding 11 applied it to spherically symmetric potentials using operator notation Suppose we can find a factorization of the Hamiltonian by operators C l displaystyle C l as C l C l 2 m H l F l displaystyle C l C l 2 mu H l F l 1 andC l C l 2 m H l 1 G l displaystyle C l C l 2 mu H l 1 G l for scalars F l displaystyle F l and G l displaystyle G l The vector C l C l C l n l displaystyle C l C l C l nl rangle may be evaluated in two different ways as C l C l C l n l 2 m E l n F l C l n l 2 m H l 1 G l C l n l displaystyle begin aligned C l C l C l nl rangle amp 2 mu E l n F l C l nl rangle amp 2 mu H l 1 G l C l nl rangle end aligned which can be re arranged as H l 1 C l n l E l n F l G l 2 m C l n l displaystyle H l 1 C l nl rangle E l n F l G l 2 mu C l nl rangle showing that C l n l displaystyle C l nl rangle is an eigenstate of H l 1 displaystyle H l 1 with eigenvalue E l 1 n E l n F l G l 2 m displaystyle E l 1 n E l n F l G l 2 mu If F l G l displaystyle F l G l then n n displaystyle n n and the states n l displaystyle nl rangle and C l n l displaystyle C l nl rangle have the same energy For the hydrogenic atom settingV r B ℏ m r displaystyle V r frac B hbar mu r with B Z m e 2 ℏ displaystyle B frac Z mu e 2 hbar a suitable equation for C l displaystyle C l is C l p r i ℏ l 1 r i B l 1 displaystyle C l p r frac i hbar l 1 r frac iB l 1 with F l G l B 2 l 1 2 displaystyle F l G l frac B 2 l 1 2 There is an upper bound to the ladder operator if the energy is negative so C l n l max 0 displaystyle C l nl max rangle 0 for some l max displaystyle l max then from Equation 1 E l n F l 2 m B 2 2 m l max 1 2 m Z 2 e 4 2 ℏ 2 l max 1 2 displaystyle E l n F l 2 mu frac B 2 2 mu l max 1 2 frac mu Z 2 e 4 2 hbar 2 l max 1 2 and n displaystyle n can be identified with l max 1 displaystyle l max 1 Relation to group theory Edit Whenever there is degeneracy in a system there is usually a related symmetry property and group The degeneracy of the energy levels for the same value of n displaystyle n but different angular momenta has been identified as the SO 4 symmetry of the spherically symmetric Coulomb potential 12 13 3D isotropic harmonic oscillator EditThe 3D isotropic harmonic oscillator has a potential given byV r m w 2 r 2 2 displaystyle V r mu omega 2 r 2 2 It can similarly be managed using the factorization method Factorization method Edit A suitable factorization is given by 11 C l p r i ℏ l 1 r i m w r displaystyle C l p r frac i hbar l 1 r i mu omega r with F l 2 l 3 m w ℏ displaystyle F l 2l 3 mu omega hbar and G l 2 l 1 m w ℏ displaystyle G l 2l 1 mu omega hbar Then E l 1 n E l n F l G l 2 m E l n w ℏ displaystyle E l 1 n E l n frac F l G l 2 mu E l n omega hbar and continuing this E l 2 n E l n 2 w ℏ E l 3 n E l n 3 w ℏ displaystyle begin aligned E l 2 n amp E l n 2 omega hbar E l 3 n amp E l n 3 omega hbar dots amp end aligned Now the Hamiltonian only has positive energy levels as can be seen from ps 2 m H l ps ps C l C l ps ps 2 l 3 m w ℏ ps C l ps C l ps 2 l 3 m w ℏ ps ps 0 displaystyle begin aligned langle psi 2 mu H l psi rangle amp langle psi C l C l psi rangle langle psi 2l 3 mu omega hbar psi rangle amp langle C l psi C l psi rangle 2l 3 mu omega hbar langle psi psi rangle amp geq 0 end aligned This means that for some value of l displaystyle l the series must terminate with C l max n l max 0 displaystyle C l max nl max rangle 0 and then E l max n F l max 2 m l max 3 2 w ℏ displaystyle E l max n F l max 2 mu l max 3 2 omega hbar This is decreasing in energy by w ℏ displaystyle omega hbar unless for some value of l displaystyle l C l n l 0 displaystyle C l nl rangle 0 Identifying this value as n displaystyle n gives E l n F l n 3 2 w ℏ displaystyle E l n F l n 3 2 omega hbar It then follows the n n 1 displaystyle n n 1 so thatC l n l l l n n 1 l 1 displaystyle C l nl rangle lambda l n n 1 l 1 rangle giving a recursion relation on l displaystyle lambda with solution l l n m w ℏ 2 n l displaystyle lambda l n mu omega hbar sqrt 2 n l There is degeneracy caused from angular momentum there is additional degeneracy caused by the oscillator potential Consider the states n n n 1 n 1 n 2 n 2 displaystyle n n rangle n 1 n 1 rangle n 2 n 2 rangle dots and apply the lowering operators C displaystyle C C n 2 n 1 n 1 C n 4 C n 3 n 2 n 2 displaystyle C n 2 n 1 n 1 rangle C n 4 C n 3 n 2 n 2 rangle dots giving the sequence n n n n 2 n n 4 displaystyle nn rangle n n 2 rangle n n 4 rangle with the same energy but with l displaystyle l decreasing by 2 In addition to the angular momentum degeneracy this gives a total degeneracy of n 1 n 2 2 displaystyle n 1 n 2 2 14 Relation to group theory Edit The degeneracies of the 3D isotropic harmonic oscillator are related to the special unitary group SU 3 14 15 History EditMany sources credit Dirac with the invention of ladder operators 16 Dirac s use of the ladder operators shows that the total angular momentum quantum number j displaystyle j needs to be a non negative half integer multiple of ħ See also EditCreation and annihilation operators Quantum harmonic oscillator Chevalley basisReferences Edit Fuchs Jurgen 1992 Affine Lie Algebras and Quantum Groups Cambridge University Press ISBN 0 521 48412 X de Lange O L R E Raab 1986 Ladder operators for orbital angular momentum American Journal of Physics 54 4 372 375 Bibcode 1986AmJPh 54 372D doi 10 1119 1 14625 Sakurai Jun J 1994 Modern Quantum Mechanics Delhi India Pearson Education Inc p 192 ISBN 81 7808 006 0 Woodgate Gordon K 1983 10 06 Elementary Atomic Structure ISBN 978 0 19 851156 4 Retrieved 2009 03 03 Angular Momentum Operators Graduate Quantum Mechanics Notes University of Virginia Retrieved 2009 04 06 David C W 1966 Ladder Operator Solution for the Hydrogen Atom Electronic Energy Levels American Journal of Physics 34 10 984 985 Bibcode 1966AmJPh 34 984D doi 10 1119 1 1972354 Burkhardt C E Levanthal J 2004 Lenz vector operations on spherical hydrogen atom eigenfunctions American Journal of Physics 72 8 1013 1016 Bibcode 2004AmJPh 72 1013B doi 10 1119 1 1758225 Pauli Wolfgang 1926 Uber das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik Z Phys 36 5 336 363 Bibcode 1926ZPhy 36 336P doi 10 1007 BF01450175 S2CID 128132824 B L Van der Waerden Sources of Quantum Mechanics Dover New York 1968 L Infeld Hull T E 1951 The Factorization Method Rev Mod Phys 23 1 21 68 Bibcode 1951RvMP 23 21I doi 10 1103 RevModPhys 23 21 a b Newmarch J D Golding R M 1978 Ladder operators for some spherically symmetric potentials in quantum Am J Phys 46 658 660 doi 10 1119 1 11225 Weinberg S J 2011 The SO 4 Symmetry of the Hydrogen Atom PDF a href Template Cite journal html title Template Cite journal cite journal a Cite journal requires journal help Lahiri A Roy P K Bagchi B 1989 Supersymmetry and the Ladder Operator Technique in Quantum Mechanics The Radial Schrodinger Equation Int J Theor Phys 28 2 183 189 Bibcode 1989IJTP 28 183L doi 10 1007 BF00669809 S2CID 123255435 a b Kirson M W 2013 Introductory Algebra for Physicists Isotropic harmonic oscillator PDF Weizmann Institute of Science Retrieved 28 July 2021 D M 33 3 1965 207 211 Fradkin D M 1965 Three dimensional isotropic harmonic oscillator and SU3 Am J Phys 33 3 207 211 Bibcode 1965AmJPh 33 207F doi 10 1119 1 1971373 https www fisica net mecanica quantica quantum harmonic oscillator lecture pdf bare URL PDF Retrieved from https en wikipedia org w index php title Ladder operator amp oldid 1134411166, wikipedia, wiki, book, books, library,