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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.

Theoretical significance and history

Quantum mechanics is concerned with discrete quanta. Angular momentum eigenstates of atoms are discrete. Raising and lowering operators serve as relations between these discrete states. Ladder operators have some practical uses as shorthand methods to generate possible states from given states. They have much deeper implications for theory because they can be used to show that discreteness of angular momentum exists even in classical physicsTemplate:Cn. They can be used to show the superiority of formal quantum mechanics over old quantum theory.

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Many sources credit Dirac with the invention of ladder operators.^[1] Dirac's use of the ladder operators goes as far as to show that the total angular momentum quantum number $l$ , or actually $j$ , needs to be a non-negative half integer multiple of ħ. He also showed how the magnetic quantum number $m$ needs to run from $- j$ to $+ j$ in integer steps of ħ. However, this is too general because $m$ is related to a rotation about the z-axis. It is non-intuitive that a full rotation does not return a system to itself (which is the reason the introduction of spin-½ was resisted at first). In order for a full rotation to return a system to itself, $m$ must be an integer multiple of ħ, not just half-integer. In order to account for this, we split the total angular momentum of the electron into the orbital component $l$ , which must obey the intuitive requirements of full rotation, and the intrinsic spin component $s$ , which is not required to do so. Once this split is made, we can easily apply physical intuition to the problem. The results, then, are in agreement with experiment.

The restriction on $l$ and $m_{l}$ to integer multiples of ħ was done by "H. E. Rorschach at the 1962 Southwestern Meeting of the American Physical Society."^[2] There is also resistance to such a split, so that, in the same year, Merzbacher^[3] derived the same boundary conditions from another angle of attack, the Aharonov-Bohm effect. The arguments and ladder operators themselves have been extended many times since, to deal with spin, and to generate more than just m for given l, but also to generate l.^[4]

Terminology

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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 a_i^† increments the number of particles in state i, while the corresponding annihilation operator a_i 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 creation/annihilation operators requires the use of an annihilation operator to remove a particle from the initial state and a creation operator to add a particle to the final state.

General formulation

Suppose that two operators X and N have the commutation relation,

[N, X] = c X,

for some scalar c. If $| n ⟩$ is an eigenstate of N with eigenvalue equation,

N | n ⟩ = n | n ⟩,

then the operator X acts on $| n ⟩$ in such a way as to shift the eigenvalue by c:

\begin{aligned} N X | n ⟩ & = (X N + [N, X]) | n ⟩ \\ = (X N + c X) | n ⟩ \\ = X N | n ⟩ + c X | n ⟩ \\ = X n | n ⟩ + c X | n ⟩ \\ = (n + c) X | n ⟩ . \end{aligned}

In other words, if $| n ⟩$ is an eigenstate of N with eigenvalue n then $X | n ⟩$ is an eigenstate of N with eigenvalue n + c. 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^{†} .

In particular, if X is a lowering operator for N then X^† is a raising operator for N and vice-versa.

Angular momentum

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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, J_x, J_y and J_z we define the two ladder operators, J₊ and J_-:^[4]

J_{+} = J_{x} + i J_{y},

J_{-} = J_{x} - i J_{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},

where ε_ijk 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 between the ladder operators and J_z can easily be obtained:

[J_{z}, J_{\pm}] = \pm ℏ J_{\pm} .

[J_{+}, J_{-}] = 2 ℏ J_{z} .

The properties of the ladder operators can be determined by observing how they modify the action of the J_z operator on a given state:

\begin{aligned} J_{z} J_{\pm} | j m ⟩ & = (J_{\pm} J_{z} + [J_{z}, J_{\pm}]) | j m ⟩ \\ = (J_{\pm} J_{z} \pm ℏ J_{\pm}) | j m ⟩ \\ = ℏ (m \pm 1) J_{\pm} | j m ⟩ . \end{aligned}

Compare this result with:

J_{z} | j m \pm 1 ⟩ = ℏ (m \pm 1) | j m \pm 1 ⟩ .

Thus we conclude that $J_{\pm} | j m ⟩$ is some scalar multiplied by $| j m \pm 1 ⟩$ ,

J_{+} | j m ⟩ = α | j m + 1 ⟩,

J_{-} | j m ⟩ = β | j m - 1 ⟩ .

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 we first take the norm of each operator, recognizing that J₊ and J_- are a Hermitian conjugate pair ( $J_{\pm} = J_{\mp}^{†}$ ),

⟨ j m | J_{+}^{†} J_{+} | j m ⟩ = ⟨ j m | J_{-} J_{+} | j m ⟩ = ⟨ j m + 1 | α^{*} α | j m + 1 ⟩ = | α |^{2}

,

⟨ j m | J_{-}^{†} J_{-} | j m ⟩ = ⟨ j m | J_{+} J_{-} | j m ⟩ = ⟨ j m - 1 | β^{*} β | j m - 1 ⟩ = | β |^{2}

.

The product of the ladder operators can be expressed in terms of the commuting pair J² and 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},

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} .

Thus we can express the values of |α|² and |β|² in terms of the eigenvalues of J² and J_z,

| α |^{2} = ℏ^{2} j (j + 1) - ℏ^{2} m^{2} - ℏ^{2} m = ℏ^{2} (j - m) (j + m + 1),

| β |^{2} = ℏ^{2} j (j + 1) - ℏ^{2} m^{2} + ℏ^{2} m = ℏ^{2} (j + m) (j - m + 1) .

The phases of α and β are not physically significant, thus they can be chosen to be real and we have:^[5]

J_{+} | j m ⟩ = ℏ \sqrt{(j - m) (j + m + 1)} | j m + 1 ⟩ = ℏ \sqrt{j (j + 1) - m (m + 1)} | j m + 1 ⟩,

J_{-} | j m ⟩ = ℏ \sqrt{(j + m) (j - m + 1)} | j m - 1 ⟩ = ℏ \sqrt{j (j + 1) - m (m - 1)} | j m - 1 ⟩ .

Confirming that m is bounded by the value of j ( $- j \leq m \leq j$ ) we have:

J_{+} | j j ⟩ = 0,

J_{-} | j - j ⟩ = 0 .

Applications in atomic and molecular physics

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,^[6]

{\hat{H}}_{D} = \hat{A} I \cdot J .

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,^[7]

\begin{aligned} J_{- 1}^{(1)} & = \frac{1}{\sqrt{2}} (J_{x} - i J_{y}) = \frac{J_{-}}{\sqrt{2}} \\ J_{0}^{(1)} & = J_{z} \\ J_{+ 1}^{(1)} & = - \frac{1}{\sqrt{2}} (J_{x} + i J_{y}) = - \frac{J_{+}}{\sqrt{2}} \end{aligned}

From these definitions it can be shown that the above scalar product can be expanded as

I^{(1)} \cdot 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 m_i = ±1 and m_j = PROPERTY builders did not have the simplest year, what with the cooling measures imposed in January and the loan curbs in June, but some still managed to do effectively while others made their first foray abroad.

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Harmonic Oscillator

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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

\begin{aligned} a & = \sqrt{\frac{m ω}{2 ℏ}} (\hat{x} + \frac{i}{m ω} \hat{p}) \\ a^{†} & = \sqrt{\frac{m ω}{2 ℏ}} (\hat{x} - \frac{i}{m ω} \hat{p}) \end{aligned}

References

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A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
↑ Template:Cite web

[WEBB-1] ttp://www.fisica.net/quantica/quantum_harmonic_oscillator_lecture.pdf

[JAMMER-2] 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[Merzbacher-3] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[Pena-4] 4.0 ^4.1 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[5] 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[Woodgate2-6] 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

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Equivalent oxide thickness

Contents

Theoretical significance and history

Terminology

General formulation

Angular momentum

Applications in atomic and molecular physics

Harmonic Oscillator

See also

References

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