Quantum energy wave function Equation and harmonic oscillator [PDF]

solution shows the possibility to electron travelling which agrees with cooper model. The critical temperature is shown

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ISSN: 2350-0328 International Journal of Advanced Research in Science, Engineering and Technology Vol. 2, Issue 12 , December 2015

Quantum energy wave function Equation and harmonic oscillator Einas Mohammed Ahmed Widaa,Mubarak Dirar Abdallah, Sawsan Ahmed Elhouri Ahmed Sudan University of Science & Technology,College of Science,Department of Physics,Khartoum,Sudan International University of Africa, College of Science,Department of Physics & Sudan University of Science & Technology-College of Science-Department of Physics-Khartoum-Sudan University of Bahri,College of Applied & Industrial Sciences,Department of Physics,Khartoum, Sudan ABSTRACT: The physics of conduction by hopping in superconductivity is not yet well established. This work is concerned with trying to throw light on conduction by hopping in superconductivity. It uses SchrΓΆdinger equation for energy wave function which time and spatial dependent or spatial dependent only. It is found that the wave function is highly localized in most cases which means that electrons conduct through hopping to adjacent atoms only. One solution shows the possibility to electron travelling which agrees with cooper model. The critical temperature is shown to depend on binding energy. KEYWORDS: Quantum energy, wave function; Harmonic oscillator. I.

INTRODUCTION

The history of quantum mechanics dates from the discovery of plank that light behaves like particles[1].Later on De Broglie proposed that particles like electrons behaves as waves[2].This confirms the dual nature of atomic and sub atomic particles[3,4].This encourages SchrΓΆdinger and Heisenberg to formulate a quantum equation that describes atomic world[5]. Heisenberg representation was developed by so called matrix representation, which represents quantum systems in different space [6].These includes energy, momentum and coordinate space. As far as the energy of atoms and electrons are important, it is there for important to study quantum SchrΓΆdinger equation in the energy space [6]. This is since the energy wave function gives the probability of electrons transition [7].This energy representation is used by the so called Hubbard model to describe superconductivity behavior [8].This model is complex and cannot directly explain why the resistance vanishes beyond some critical temperature [9].How ever some attempts were model to do this [9].But this model is mathematically complex. Thus there is a need for simple model. This is done in this work. Section 2 is concerned with energy wave function equation for spatially and time dependent case. Harmonic oscillator solution is in section 3. Section 4 is devoted for spatial dependent wave function and particle in a box solution discontinuity and can clear in section 5 and 6 respectively.

II.QUANTUM ENERGY WAVE FUNCTION It is very investing to see how the energy wave function can evolve with time and coordinates . ThusCk Is a function of x and t: |𝛹>= βˆ‘ck|uk>(1) Where : πΆπ‘˜= π‘π‘˜ π‘₯ , 𝑑 (2) In this case SchrΓΆdinger equation becomes: Ə|𝛹> iΡ›= = Δ€|𝛹>(3) Ə

Ə𝑑

iΡ› βˆ‘kck|uk> = βˆ‘k Δ€ck|uk> Ə𝑑 πœ•

iΡ› π‘π‘˜ π‘’π‘˜ = π‘˜ 𝐻 π‘π‘˜ π‘’π‘˜ = π‘˜ (𝐻0+ 𝑉1 )π‘π‘˜ π‘’π‘˜ πœ•π‘‘ Where the perturbed Hamiltonian: Copyright to IJARSET

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ISSN: 2350-0328 International Journal of Advanced Research in Science, Engineering and Technology Vol. 2, Issue 12 , December 2015

𝐻 = 𝐻0 + 𝑉1 Thus: πœ•π‘

βˆ‚u

𝑖ћ π‘˜ π‘’π‘˜ + 𝑖ћck ( k = π‘˜ 𝐻 π‘π‘˜ π‘’π‘˜ πœ•π‘‘ βˆ‚t But for non perturbed system [see equation (1)]:

(4)

Ρ›2

πœ•π‘’

iΡ› π‘˜ =𝐸k uk= 𝐻0 π‘’π‘˜ = βˆ’ βˆ‡2 π‘’π‘˜ + 𝑉0 π‘’π‘˜ πœ•π‘‘ 2π‘š Thus:

(5) Ρ›2

βˆ‚c

iΡ› k uk + ck Ek uk = π‘˜ Hck uk +𝛻 2 π‘π‘˜ uk π‘˜ βˆ’ = βˆ‚t 2m 𝑉0 𝑐𝐾 π‘’π‘˜ + 𝑉1 π‘π‘˜ π‘’π‘˜ (6) But the second term in the right hand side of(6) is given by: Ρ›2 βˆ’ u βˆ‡2 ck + ck βˆ‡2 uk + 2βˆ‡uk βˆ‡ck 2 βˆ‡ π‘π‘˜ uk + Vck uk = 2m k +V0 ck uk + V1 ck uk k π‘˜

V1 uk ck + From(5): π‘˜

π‘˜

βˆ’

Ρ›2 2m

βˆ‡2 uk + V0 uk ck + =

E k uk c k + π‘˜

k

βˆ’ k

βˆ’

Ρ›2 2m

βˆ’ π‘˜

Ρ›2 2m

uk βˆ‡2 ck + 2βˆ‡uk βˆ‡ck

Ρ›2 u βˆ‡2 ck+ 2βˆ‡uk βˆ‡ck + 2m k

V1 uk ck

(7)

k

Thus from(6): Ρ›2

βˆ‚c

Ρ›2

ih k + βˆ‡2 ck uk = βˆ’ βˆ‚t 2m m Setting: βˆ‡π‘k α΄ͺ = And using the hermiticity of𝑝: π‘˜

k

βˆ‡uk βˆ‡ck +

π‘˜ 𝑉1 π‘’π‘˜ π‘π‘˜

(8)

𝑝𝑒 α΄ͺπ‘‘π‘Ÿ =

π‘’π‘˜ 𝑝 α΄ͺπ‘‘π‘Ÿ

π‘˜

βˆ’Ρ› i π‘’π‘˜ βˆ‡2 π‘π‘˜ π‘‘π‘Ÿ-(9)

π›»π‘’π‘˜ βˆ‡π‘π‘˜ π‘‘π‘Ÿ = Thus: βˆ‡π‘’π‘˜ π›»π‘π‘˜ =βˆ’π‘’ π‘˜ βˆ‡2 π‘π‘˜ (10) Substituting (10)in(8) yields: πœ•π‘π‘˜ Ρ›2 2 𝑖ћ βˆ’ βˆ‡ π‘π‘˜ π‘’π‘˜= πœ•π‘‘ 2π‘š π‘˜

𝛻 uk α΄ͺπ‘‘π‘Ÿ = βˆ’

Ρ› i

π‘’π‘˜ βˆ‡α΄ͺπ‘‘π‘Ÿ

𝑉1 π‘’π‘˜ π‘π‘˜ 11 π‘˜

βˆ‚ck Ρ›2 2 iΡ› βˆ’ βˆ‡ ck βˆ’ V1 ck 𝑒k = 0 βˆ‚t 2m

π‘˜

Can be solving by selecting, this: iΡ›

πœ•π‘ π‘˜ πœ•π‘‘

βˆ’

Ρ›2 2π‘š

βˆ‡2 π‘π‘˜ βˆ’ 𝑉1 π‘π‘˜ = 0

(12) III.

Harmonic oscillator

Consider equation (12) : Ə𝐢

Ρ›Β²

iΡ› π‘˜ = βˆ‡Β²π‘π‘˜ + 𝑉1 π‘π‘˜ (13) Ə𝑑 2π‘š For time independent potential, let: π‘π‘˜ =𝑒 +𝑖 / Ρ›πΈπ‘˜ t π‘’π‘˜ (14) To get: πΈπ‘˜ π‘’π‘˜ =

Ρ›Β² 2π‘š

βˆ‡Β²π‘’π‘˜ +𝑉1 π‘’π‘˜ (15)

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ISSN: 2350-0328 International Journal of Advanced Research in Science, Engineering and Technology Vol. 2, Issue 12 , December 2015

For harmonic oscillator perturbation by electromagnetic field : 𝑉1 = Β½ kπ‘₯ 2 = 𝑐1 xΒ² (16) Thus: Ρ›Β² 2m

βˆ‡Β²uk + c1 xΒ² =-Ek ck

(17)

Substituting: π‘’π‘˜ = A 𝑒 βˆ’π›Ό π‘₯Β² (18) βˆ‡π‘’π‘˜ =-2 𝛼 x 𝑒 βˆ’ 𝛼 π‘₯ Β² = - 2 𝛼 x π‘’π‘˜ βˆ‡Β²π‘’π‘˜ = -2 π›Όπ‘’π‘˜ - 2 𝛼x βˆ‡π‘’π‘˜ = -2 π›Όπ‘’π‘˜ + 4 𝛼² xΒ²π‘’π‘˜ (19) Ρ›Β² [ -2 𝛼 + 4 𝛼²xΒ²+ 𝑐1 xΒ² ] π‘’π‘˜ =βˆ’πΈπ‘˜ π‘’π‘˜ (20) 2π‘š πΈπ‘˜ =

Ρ›Β² π‘š

𝛼 ; 4𝛼² = - 𝑐1 (23) 𝑖

𝑖 π‘š

k = 2 2Ο‰ Thus from (14) and (18) beside (23): 𝛼 = Β½ βˆ’ π‘˜/2 = 𝑖𝐸 π‘˜ 𝑇

π‘–π‘˜

2

π‘₯

2

π‘π‘˜ = A 𝑒 + Ρ› 𝑒 2 2 (24) Which represents non localized travelling wave? However for particles affected by additional perturbing potentials like applying sound wave with frequency πœ”π‘  and electron magnetic wave with frequencyπœ”π‘’ , such that the two forces apposes each other; in this case : - βˆ‡ 𝑉 = F = m π‘₯ = 𝐹𝑒 βˆ’ 𝐹𝑠 = - π‘˜π‘’ x + π‘˜π‘  x Ə𝑉 = - m πœ”π‘’ x + m πœ” 𝑠 x Ə𝑋 Thus : V = Β½ mπœ”π‘  2 xΒ² - Β½ m πœ”π‘’ 2 xΒ² = Β½ m(πœ”π‘  2 - πœ”π‘’ 2 ) xΒ² =1/2m(πœ”π‘  + πœ”π‘’ )(πœ”π‘  βˆ’ πœ”π‘’ )π‘₯ 2 = - Β½ k xΒ² K = m (πœ”π‘  + πœ”π‘’ ) (πœ” 𝑒 βˆ’πœ”π‘  ) = 𝐢1 (25) When : πœ”π‘  >πœ”π‘’ In this case : 𝐢2 = - 𝐢1 = m (πœ”π‘  + πœ”π‘’ ) (πœ”π‘  βˆ’ πœ”π‘’ ) > 0 Thus equation (23) gives : Ξ± = Β½ 𝑐2 (27) Due to the periodicity of πΆπ‘˜ in (14) : πΆπ‘˜ ( t + T ) = πΆπ‘˜ (t) Thus :

(26)

+𝑖

𝑒 Ρ› πΈπ‘˜ T = 1 𝐸 𝐸 Cos π‘˜ 𝑇 = 1 ; sin π‘˜ 𝑇 Ρ› Ρ› Hence : T Ek =2nπœ‹ Ρ›

=0

(28)

2Ο€

Ek = n Ρ› ( ) = n Ρ›Ο‰ T Thus from equation ( 23) : m n mΟ‰ Ξ± = Ek = (29) Ρ›Β² Ρ› From equation(18): π‘’π‘˜ = A 𝑒 βˆ’π›Ό π‘₯Β² Thus the energy is quantized, and is mighty localized thus move by hopping.

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ISSN: 2350-0328 International Journal of Advanced Research in Science, Engineering and Technology Vol. 2, Issue 12 , December 2015

IV. Spatial dependent energy wave function and particles in a box SchrΓΆdinger equation which i s based on Newtonian energy relation is gives by : Ə𝛹

Ρ›Β²

iΡ› = = - βˆ‡Β²π›Ή + V 𝛹 ( 30) Ə𝑑 2π‘š To make use of equation (30) consider: π‘π‘˜ = π‘π‘˜ ( E , x) (31) i.e , the energy wave function depends on E and x only , where : π‘’π‘˜ = π‘’π‘˜ ( E , t) = A 𝑒 𝑖 πΈπ‘˜ 𝑑 / Ρ› (32) substituting equation ( 31) in (30) yields : Ə π‘’π‘˜ βˆ’ Ρ›Β² π‘˜ π‘π‘˜ Ə𝑑 = 2π‘š Ə π‘’π‘˜ βˆ’ Ρ›Β² π‘˜ π‘π‘˜ Ə𝑑 = 2π‘š

iΡ›

π›»Β²π‘π‘˜ 𝑒𝑒 +

π›»Β²πΆπ‘˜ π‘’π‘˜ + In view of equation : Ə𝑒 iΡ› π‘˜ = πΈπ‘˜ π‘’π‘˜ Ə𝑑 Thus :

π‘˜ Vπ‘’π‘˜ πΆπ‘˜ π‘˜

(33)

V π‘’π‘˜ πΆπ‘˜ (34)

βˆ’ Ρ›Β²

π‘˜ πΈπ‘˜ πΆπ‘˜

π‘’π‘˜ = π‘˜ π‘’π‘˜ π›»Β²π‘π‘˜ + π‘˜ V π‘’π‘˜ π‘π‘˜ (35) 2π‘š Multiply equation (35) by 𝑒𝑛 , then integrating , yields : βˆ’Ρ› π‘˜ πΈπ‘˜ π‘π‘˜ π‘’π‘˜ 𝑒𝑛 .dr = π‘˜ ( π‘’π‘˜ 𝑒𝑛 .dr ) π›»Β²π‘π‘˜ + π‘˜ π‘π‘˜ 𝑒𝑛 V π‘’π‘˜ π‘‘π‘Ÿ βˆ’Ρ›

π‘˜ πΈπ‘˜ π‘π‘˜

2π‘š

𝛿 π‘›π‘˜ = (π›»Β²π‘π‘˜ ) π›Ώπ‘›π‘˜ + π‘˜ π‘π‘˜ π‘‰π‘›π‘˜ 2π‘š π‘˜ π›Ώπ‘›π‘˜ = 𝛿𝑛𝑛 = 1 ( n =k) Thus schordinger equation for energy wave function is give by : βˆ’ Ρ›Β²

𝑐𝑛 𝐸𝑛 = 𝛻²𝑐𝑛 + π‘˜ π‘π‘˜ π‘‰π‘›π‘˜ (36) 2π‘š For constant potential 𝑉0 equation (36) reduces to : 𝑐𝑛 𝐸𝑛 = Thus :

βˆ’ Ρ›Β² 2π‘š

𝛻²𝑐𝑛 +

π‘˜ π‘π‘˜ 𝑉0 π›Ώπ‘˜π‘›

βˆ’ Ρ›Β²

𝛻²𝑐𝑛 = (𝐸𝑛 βˆ’ 𝑉0 ) 𝑐𝑛 (37) To solve this equation consider the solution of equation(37) to be : 𝑐𝑛 = A 𝑒 π‘–π‘˜π‘₯ (38) Thus : K = 2π‘š(𝐸𝑛 βˆ’π‘‰0 ) (39) For highly localized electrons : 𝐸𝑛

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