Condensation Happens Without Any Attraction: What Decides Is the Wavelength Overtaking the Spacing ── the condensation temperature of an ideal Bose gas contains only constants, the mass and the number density, with no interparticle potential at all ── the separator is whether switching off the interaction makes it disappear ── [Paper 663]
It is natural to think that a phase changes because particles attract one another. But the condensation of bosons happens without any attraction at all. The condensation temperature of an ideal Bose gas is computed in three cases, and what fixes that temperature is examined. No new theorem or law is claimed. Scope of this paper (scope note): No new theorem or law is claimed──condensation of the ideal Bose gas (Einstein 1925), the thermal de Broglie wavelength, and the value of the Riemann zeta function at 3/2 are all standard. Interacting condensates are not treated──scattering lengths, mean-field equations and the shift of the condensation temperature by interactions are not included. Only the ideal gas is examined. Experiments are not discussed──how the gas is cooled (evaporative cooling, magneto-optical traps) and how the condensate is observed are not the subject. Superconductivity is not explained──condensation that requires pairing is not entered into. The separator "does it disappear when the interaction is switched off" is placed, and no more. Finite systems are not treated──the counting is in the thermodynamic limit, and the rounding that comes with small particle numbers is not included. Relation to earlier papers: Paper 482 showed that the words "critical dimension" name two different numbers above and below, and stated that it did not treat condensation itself, the existence proof of a phase transition, behaviour in finite systems, or the interacting case──this paper takes that seat and counts, for the same ideal Bose gas, the condition for condensation. What is added is showing from the contents of the formula that no interaction enters the condensation temperature, confirming to 10 digits that three different cases return the same phase-space density, deriving the ratio of wavelength to mean spacing, and placing the separator on switching off the interaction. First, no interparticle potential appears in the formula for the condensation temperature──only the Planck constant, the Boltzmann constant, the mass of the particle and the number density enter it; no quantity measuring the strength of an interaction appears (Section 2). Second, the temperature is nevertheless definite──Rb87 at 10^20 per cubic metre gives 398.3307 nK, and at the density of liquid helium (2.181607x10^28 particles) 3.134378 K, the latter being 1.439902 times the measured lambda point of 2.1768 K (Section 2). Third, and this is the core. The condition is a single dimensionless number──at the condensation temperature, the number density times the cube of the thermal de Broglie wavelength is exactly 2.6123753487, agreeing to 10 digits across three cases of different mass and density (Section 3). Fourth, the boundary is where the wavelength overtakes the spacing──for Rb87 the mean spacing is 2.154435x10^-7 m while the thermal de Broglie wavelength at the condensation temperature is 2.967189x10^-7 m, or 1.377247 times as long (Section 3). Fifth, raising the density does not raise the temperature in step──ten times the density gives only 4.641589 times the temperature, which is ten to the two thirds (Section 2). Sixth, the separator is whether switching off the interaction makes it disappear──the computation here contains no interaction and still condenses, so switching one off changes nothing. A condensation that requires pairs to form does disappear (Section 4). Condensation does not happen because particles attract one another. The formula for the condensation temperature of an ideal Bose gas contains only constants, the mass and the number density, and it still returned definite temperatures: 398.3307 nK for Rb87 and 3.134378 K at the density of liquid helium. At those temperatures, three cases of widely different mass and density returned the same phase-space density, 2.6123753487──the condition is not the strength of an attraction but the thermal de Broglie wavelength overtaking the spacing between particles, by a factor of 1.377247 in the example here. The separator is whether switching off the interaction makes it disappear. Placed among the earlier papers──in the seat Paper 482 left open by not treating condensation itself, the condition for it was counted. To be honest──the factor of 1.439902 against the lambda point is the work of interactions, and its size is not explained here. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. Keywords: Bose-Einstein condensation, ideal Bose gas, phase-space density, thermal de Broglie wavelength, critical temperature, Bose statistics. ----- 相が変わるのは、粒子どうしが引き合うからだと考えるのは自然である。だがボース粒子の凝縮は、引力を一つも入れなくても起きる。理想ボース気体の凝縮温度を三つの場合で数え、何がその温度を決めているのかを見る。新しい定理も法則も主張しない。 本稿の射程(射程注記):新しい定理も法則も主張しない──理想ボース気体の凝縮(アインシュタイン 1925)、熱的ド・ブロイ波長、リーマンのゼータ関数の 3/2 での値は、いずれも既知である。相互作用のある凝縮体を扱わない──散乱長、平均場の方程式、相互作用による凝縮温度のずれは入れない。理想気体だけを見る。実験を論じない──どう冷やすか(蒸発冷却、磁気光学トラップ)、どう観測するかは主題ではない。超伝導を説明しない──対をつくる凝縮に本稿は立ち入らない。分離子として「相互作用を切ったら消えるか」を置くにとどめる。有限の系を扱わない──熱力学極限で数えており、粒子数が少ないときの丸みは入れない。既刊との関係:論文482 は「臨界次元」という一語が上と下で別の数を指すことを示し、射程注記で「凝縮そのものを論じない──相転移の存在証明、有限系での振舞い、相互作用のある場合は主題ではない」と明記した──本稿はその席に座り、同じ理想ボース気体で、凝縮の条件そのものを数える。加えたのは、凝縮温度の式に相互作用が入っていないことを式の中身で示したこと、三つの場合で位相空間密度が同じ定数になることを 10 桁で確かめたこと、波長と粒子間距離の比を出したこと、分離子を「相互作用を切ったら消えるか」に置いたことである。 第一に、凝縮温度の式には、粒子間のポテンシャルが一つも入っていない──入っているのはプランク定数、ボルツマン定数、粒子の質量、数密度の四つだけで、相互作用の強さを表す量は現れない(第2節)。 第二に、それでも温度は具体的に決まる──Rb87 を 1 立方メートルあたり 10^20 個にすると 398.3307 nK、液体ヘリウムの密度(2.181607x10^28 個)なら 3.134378 K で、後者は実測のラムダ点 2.1768 K の 1.439902 倍だった(第2節)。 第三に、これが本稿の芯である。条件は一つの無次元の数である──凝縮温度では、数密度に熱的ド・ブロイ波長の三乗を掛けた量がちょうど 2.6123753487 になり、質量も密度も違う三つの場合で 10 桁一致した(第3節)。 第四に、波長が粒子の間隔を越えるところが境目である──Rb87 の例では、粒子間の平均距離 2.154435x10^-7 m に対して、凝縮温度での熱的ド・ブロイ波長は 2.967189x10^-7 m で、1.377247 倍だった(第3節)。 第五に、密度を上げても温度は同じようには上がらない──密度を 10 倍にしても凝縮温度は 4.641589 倍にしかならず、これは 10 の 2/3 乗である(第2節)。 第六に、分離子は、相互作用を切ったら消えるかである──本稿の計算は相互作用を一つも含まずに凝縮を出しており、切っても消えない。対をつくる必要がある凝縮は、切れば消える(第4節)。 凝縮は、粒子どうしが引き合うから起きるのではない。理想ボース気体の凝縮温度の式に入っているのは定数と質量と数密度だけで、それでも Rb87 で 398.3307 nK、液体ヘリウムの密度で 3.134378 K という具体的な温度が出た。その温度では、質量も密度も大きく違う三つの場合が、同じ位相空間密度 2.6123753487 を返した──条件は引力の強さではなく、熱的ド・ブロイ波長が粒子間の距離を越えること(本稿の例では 1.377247 倍)である。分離子は、相互作用を切ったら消えるかである。既刊との位置──論文482 が「凝縮そのものは論じない」と空けた席で、その条件を数えた。正直に言えば──ラムダ点との 1.439902 倍の差は相互作用の効きであり、本稿はその大きさを説明していない。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。 キーワード:ボース・アインシュタイン凝縮、理想ボース気体、位相空間密度、熱的ド・ブロイ波長、臨界温度、ボース統計。
It is natural to think that a phase changes because particles attract one another. But the condensation of bosons happens without any attraction at all. The condensation temperature of an ideal Bose gas is computed in three cases, and what fixes that temperature is examined. No new theorem or law is claimed. Scope of this paper (scope note): No new theorem or law is claimed──condensation of the ideal Bose gas (Einstein 1925), the thermal de Broglie wavelength, and the value of the Riemann zeta function at 3/2 are all standard. Interacting condensates are not treated──scattering lengths, mean-field equations and the shift of the condensation temperature by interactions are not included. Only the ideal gas is examined. Experiments are not discussed──how the gas is cooled (evaporative cooling, magneto-optical traps) and how the condensate is observed are not the subject. Superconductivity is not explained──condensation that requires pairing is not entered into. The separator "does it disappear when the interaction is switched off" is placed, and no more. Finite systems are not treated──the counting is in the thermodynamic limit, and the rounding that comes with small particle numbers is not included. Relation to earlier papers: Paper 482 showed that the words "critical dimension" name two different numbers above and below, and stated that it did not treat condensation itself, the existence proof of a phase transition, behaviour in finite systems, or the interacting case──this paper takes that seat and counts, for the same ideal Bose gas, the condition for condensation. What is added is showing from the contents of the formula that no interaction enters the condensation temperature, confirming to 10 digits that three different cases return the same phase-space density, deriving the ratio of wavelength to mean spacing, and placing the separator on switching off the interaction. First, no interparticle potential appears in the formula for the condensation temperature──only the Planck constant, the Boltzmann constant, the mass of the particle and the number density enter it; no quantity measuring the strength of an interaction appears (Section 2). Second, the temperature is nevertheless definite──Rb87 at 10^20 per cubic metre gives 398.3307 nK, and at the density of liquid helium (2.181607x10^28 particles) 3.134378 K, the latter being 1.439902 times the measured lambda point of 2.1768 K (Section 2). Third, and this is the core. The condition is a single dimensionless number──at the condensation temperature, the number density times the cube of the thermal de Broglie wavelength is exactly 2.6123753487, agreeing to 10 digits across three cases of different mass and density (Section 3). Fourth, the boundary is where the wavelength overtakes the spacing──for Rb87 the mean spacing is 2.154435x10^-7 m while the thermal de Broglie wavelength at the condensation temperature is 2.967189x10^-7 m, or 1.377247 times as long (Section 3). Fifth, raising the density does not raise the temperature in step──ten times the density gives only 4.641589 times the temperature, which is ten to the two thirds (Section 2). Sixth, the separator is whether switching off the interaction makes it disappear──the computation here contains no interaction and still condenses, so switching one off changes nothing. A condensation that requires pairs to form does disappear (Section 4). Condensation does not happen because particles attract one another. The formula for the condensation temperature of an ideal Bose gas contains only constants, the mass and the number density, and it still returned definite temperatures: 398.3307 nK for Rb87 and 3.134378 K at the density of liquid helium. At those temperatures, three cases of widely different mass and density returned the same phase-space density, 2.6123753487──the condition is not the strength of an attraction but the thermal de Broglie wavelength overtaking the spacing between particles, by a factor of 1.377247 in the example here. The separator is whether switching off the interaction makes it disappear. Placed among the earlier papers──in the seat Paper 482 left open by not treating condensation itself, the condition for it was counted. To be honest──the factor of 1.439902 against the lambda point is the work of interactions, and its size is not explained here. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. Keywords: Bose-Einstein condensation, ideal Bose gas, phase-space density, thermal de Broglie wavelength, critical temperature, Bose statistics. ----- 相が変わるのは、粒子どうしが引き合うからだと考えるのは自然である。だがボース粒子の凝縮は、引力を一つも入れなくても起きる。理想ボース気体の凝縮温度を三つの場合で数え、何がその温度を決めているのかを見る。新しい定理も法則も主張しない。 本稿の射程(射程注記):新しい定理も法則も主張しない──理想ボース気体の凝縮(アインシュタイン 1925)、熱的ド・ブロイ波長、リーマンのゼータ関数の 3/2 での値は、いずれも既知である。相互作用のある凝縮体を扱わない──散乱長、平均場の方程式、相互作用による凝縮温度のずれは入れない。理想気体だけを見る。実験を論じない──どう冷やすか(蒸発冷却、磁気光学トラップ)、どう観測するかは主題ではない。超伝導を説明しない──対をつくる凝縮に本稿は立ち入らない。分離子として「相互作用を切ったら消えるか」を置くにとどめる。有限の系を扱わない──熱力学極限で数えており、粒子数が少ないときの丸みは入れない。既刊との関係:論文482 は「臨界次元」という一語が上と下で別の数を指すことを示し、射程注記で「凝縮そのものを論じない──相転移の存在証明、有限系での振舞い、相互作用のある場合は主題ではない」と明記した──本稿はその席に座り、同じ理想ボース気体で、凝縮の条件そのものを数える。加えたのは、凝縮温度の式に相互作用が入っていないことを式の中身で示したこと、三つの場合で位相空間密度が同じ定数になることを 10 桁で確かめたこと、波長と粒子間距離の比を出したこと、分離子を「相互作用を切ったら消えるか」に置いたことである。 第一に、凝縮温度の式には、粒子間のポテンシャルが一つも入っていない──入っているのはプランク定数、ボルツマン定数、粒子の質量、数密度の四つだけで、相互作用の強さを表す量は現れない(第2節)。 第二に、それでも温度は具体的に決まる──Rb87 を 1 立方メートルあたり 10^20 個にすると 398.3307 nK、液体ヘリウムの密度(2.181607x10^28 個)なら 3.134378 K で、後者は実測のラムダ点 2.1768 K の 1.439902 倍だった(第2節)。 第三に、これが本稿の芯である。条件は一つの無次元の数である──凝縮温度では、数密度に熱的ド・ブロイ波長の三乗を掛けた量がちょうど 2.6123753487 になり、質量も密度も違う三つの場合で 10 桁一致した(第3節)。 第四に、波長が粒子の間隔を越えるところが境目である──Rb87 の例では、粒子間の平均距離 2.154435x10^-7 m に対して、凝縮温度での熱的ド・ブロイ波長は 2.967189x10^-7 m で、1.377247 倍だった(第3節)。 第五に、密度を上げても温度は同じようには上がらない──密度を 10 倍にしても凝縮温度は 4.641589 倍にしかならず、これは 10 の 2/3 乗である(第2節)。 第六に、分離子は、相互作用を切ったら消えるかである──本稿の計算は相互作用を一つも含まずに凝縮を出しており、切っても消えない。対をつくる必要がある凝縮は、切れば消える(第4節)。 凝縮は、粒子どうしが引き合うから起きるのではない。理想ボース気体の凝縮温度の式に入っているのは定数と質量と数密度だけで、それでも Rb87 で 398.3307 nK、液体ヘリウムの密度で 3.134378 K という具体的な温度が出た。その温度では、質量も密度も大きく違う三つの場合が、同じ位相空間密度 2.6123753487 を返した──条件は引力の強さではなく、熱的ド・ブロイ波長が粒子間の距離を越えること(本稿の例では 1.377247 倍)である。分離子は、相互作用を切ったら消えるかである。既刊との位置──論文482 が「凝縮そのものは論じない」と空けた席で、その条件を数えた。正直に言えば──ラムダ点との 1.439902 倍の差は相互作用の効きであり、本稿はその大きさを説明していない。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。 キーワード:ボース・アインシュタイン凝縮、理想ボース気体、位相空間密度、熱的ド・ブロイ波長、臨界温度、ボース統計。
Abstract Einstein used double quotes on two words, because in the concept of an ideal gas there is no interaction between its constituents, whereas condensation and saturation are properties of a real gas of interacting atoms or molecules. The ‘interaction’ bringing about Bose–Einstein condensation is the symmetry requirement for multi-particle wave functions of bosons, sometimes called exchange force (see page 176). For this to become effective, the wave functions of individual bosons have to overlap, i.e., the average (or thermal ) de Broglie wavelength of the bosons has to be on the order of or larger than the average distance between them. In other words, a Bose–Einstein condensate is expected to form if the ratio of wavelength and distance exceeds a certain value. (The third power of this ratio is called phase-space density, see margin.) At very low temperature the average distance can be quite large; interactions between bosons become unimportant and the situation of an ideal gas is well approximated. We saw in Episode 97 how such temperatures for samples of neutral atoms were reached. These methods, greatly simplified and extended, were the basis for the achievement of Bose–Einstein condensates by the groups ofWieman and Cornell [2] and of Ketterle [3] in 1995, seventy years after Einstein’s prediction.
Quantum systems consisting of many particles are both interesting from the physics and the mathematics perspective. One aspect that makes them fascinating from the physics point of view is that they can exhibit quantum effects on macroscopic scales such as Bose-Einstein condensation. This phenomenon, first predicted 100 years ago by Bose [19] and Einstein [44], occurs when a dilute gas of bosons is cooled down to very low temperatures. Below a certain critical temperature a condensate forms, meaning that a macroscopic fraction of the particles is in the same one-particle state, and thus, quantum phenomena become relevant on length scales much larger than the individual atom. The occurrence of Bose-Einstein condensation is also related to superfluidity as suggested in [88]. Bose-Einstein condensation has been achieved experimentally independently by the groups of Cornell and Wieman [3] and Ketterle [40] earning them a Nobel prize in 2001. While Bose and Einstein analyzed theoretically the non-interacting gas, an effective theory describing weakly interacting Bose gases was proposed by Bogoliubov in 1947 with the goal of explaining the occurrence of superfluidity in liquid helium [18]. Using assumptions such as that condensation indeed occurs, his theory yields predictions for the low-energy states of the system. Seen from the mathematics perspective, describing the Bose gas is an interesting challenge due to the strong correlations between particles introduced by the interaction making this case much harder to study than the non-interacting one. In particular, a proof that Bose-Einstein condensation occurs in the thermodynamic limit for continuous systems is still out of reach. However, there has been a lot of work regarding the rigorous analysis of bosonic systems in various relevant regimes. This thesis contains three new contributions establishing rigorous results on bosonic systems in different, physically relevant settings. It is based on the articles [37, 27, 31]. The first result concerns the time-evolution of initially trapped Bose gases in the ultra-dilute Gross-Pitaevskii regime. Under some assumptions on the initial data satisfied in particular by a class of states that are expected to approximate the ground state in a trap, we derive a quasi-free approximation that is close to the true solution in L2-norm. In particular, we show that this approximate wave function is described in terms of a Bose-Einstein condensate in a one-particle state following the time-dependent Gross-Pitaevskii equation together with pair excitations with respect to this condensate. As a corollary, we establish a central limit theorem for bounded one-particle observables. The second result is a simple proof of Bose-Einstein condensation for interacting, ultra-dilute gases. More specifically, we consider scalings interpolating between the Gross-Pitaevskii regime and the thermodynamic limit. While condensation had been shown to hold in these regimes before, the advantage of the method used here is its simplicity. Lastly, the third result establishes an upper bound on the ground state energy of a dilute Bose gas in the thermodynamic limit. It captures rigorously the third order correction to the energy per unit volume for sufficiently low densities as predicted in the physics literature first by Wu and subsequently others. Full-text embargoed until: 2026-01-01
From a certain temperature on, the molecules “condense” without attractive forces; that is, they accumulate at zero velocity. The theory is pretty, but is there also some truth to it? Albert Einstein, Letter to Ehrenfest (Dec. 1924). Abraham Pais, Subtle Is the Lord: The Science and the Life of Albert Einstein , Oxford University Press, New York (1982) Introduction For over 50 years the low-temperature liquid state of uncharged, spinless He 4 was the only system in which a Bose–Einstein (BE) condensation was considered experimentally realized. In that case, cold (<2.19K) liquid He 4 passes into an extraordinary phase of matter called a superfluid , in which the liquid's viscosity and entropy become zero. With advances in atomic cooling (to ≈ 10 –9 K) the number of Bose systems which demonstrably pass into a condensate has considerably increased. These include several isotopes of alkali gas atoms as well as fermionic atoms that pair into integerspin (boson) composites. Although an ideal Bose gas does exhibit a low-temperature critical instability, the ideal BE gas theory is not, on its own, able to describe the BE condensate wave state. In order for a theory of bosons to account for a condensate wave state, interactions between the bosons must be included. Nevertheless, considerable interesting physics is contained in the ideal Bose gas model.
Bose-Einstein condensation is a unique phase transition in that it is not driven by interparticle interactions, but can theoretically occur in an ideal gas, purely as a consequence of quantum statistics. This chapter addresses the question, ‘ How is this ideal Bose gas condensation modified in the presence of interactions between the particles?’ This seemingly simple question turns out to be surprisingly difficult to answer. Here we outline the theoretical background to this question and discuss some recent measurements on ultracold atomic Bose gases that have sought to provide some answers. Introduction Unlike the vast majority of phase transitions, Bose-Einstein condensation (BEC) is not driven by interparticle interactions but can theoretically occur in an ideal (noninteracting) gas, purely as a consequence of quantum statistics. However, in reality, interactions are needed for a Bose gas to remain close to thermal equilibrium. It is thus interesting to discuss if something close to ideal gas BEC can be observed in a real system and what happens in the vicinity of the BEC transition in the presence of interparticle interactions. These simple questions have not been easy to answer, either theoretically or experimentally. The theoretical foundations for studying the effect of interactions on Bose condensed systems were laid over half a century ago by Bogoliubov [1], Penrose and Onsager [2], and Belieav [3], among others. These works initially focused on zerotemperature properties and were extended to nonzero temperature in the pioneering papers of Lee, Huang, and Yang [4, 5, 6, 7]. At that time, the main experimental system was liquid He-4 in which the inter-particle interactions are strong, making connections with theory difficult. The realisation in 1995 of BEC in weakly interacting ultracold atomic gases [8, 9] thus opened up the possibility to experimentally revisit some of these long-discussed questions. This was further aided by, among other advances, the use of Feshbach resonances to tune the interaction strength in atomic gases [10, 11]. Thus, in the last twenty years the study of ultracold Bose gases has been very successful (see Chapter 3 for an insightful account of selected topics and surprising developments over this exciting period). However, the fact that, until recently, ultracold atoms were confined using harmonic potentials has hindered the study of the BEC transition itself.
Abstract The Bose-Einstein condensation of a gas is investigated. Starting from the well-known formulae for Bose statistics, the problem has been generalized to include a variety of potential fields in which the particles of the gas move, and the number w of dimensions has not been restricted to three. The energy levels are taken to be εi≡εs1,....,s10=constanth2ms1α−1a12+...+swαaw2(1≤α≤2) the quantum numbers being s1 ,w= 1, 2, ..., and a1, ..., aw being certain characteristic lengths. (For α = 2, the potential field is that of the w-dimensional rectangular box; for α = 1, we obtain the w-dimensional harmonic oscillator field.) A direct rigorous method is used similar to that proposed by Fowler & Jones (1938). It is shown that the number q = w/α determines the appearance of an Einstein transition temperature T0·For q≤ 1 there is no such point, while for q > 1 a transition point exists. For 1 < q≤ 2, the mean energy ϵ- per particle and the specific heat dϵ-/dT are continuous at T = T0· For q > 2, the specific heat is discontinuous at T = T0, giving rise to a A λ-point. A well-defined transition point only appears for a very large (theoretically infinite) number N of particles. T0 is finite only if the quantity v = N/(a1 .... aw)2/ α¯is finite. For a rectangular box, v is equal to the mean density of the gas. If v tends to zero or infinity as N→ ∞, then T0 likewise tends to zero or infinity. In the case q > 1, and at temperatures T < T 0' there is a finite fraction N0 /N of the particles, given by N 0/N = 1-(T/T0)q, in the lowest state. London’s formula (1938 b) for the three-dimensional box is an example of this equation. Some further results are also compared with those given by London’s continuous spectrum approximation.
An exact expression for the single-particle reduced density matrix of a translationally invariant system of bosons is obtained, as an average of a functional of the density and current operators. An approximation scheme, based on small fluctuations in density and current, is used in its lowest order, to approximate the single-particle reduced density matrix and the condensate density by a functional of the structure factor, which is a measurable quantity. This is done for the ground state and for finite temperatures. The resulting expressions for the condensate density are discussed. For the ideal gas the result is correct below the condensation temperature. For the case of an interacting liquid, Bose-Einstein condensation exists, in the ground states for two- and three-dimensional systems, and for finite temperatures it does exist only in three dimensions. The present experimental knowledge of the structure factor enables only a crude estimate of the condensed fraction which is found to be 0.1-0.2.